NBER WORKING PAPER SERIES

CAN’T WE ALL BE MORE LIKE SCANDINAVIANS? ASYMMETRIC GROWTHAND INSTITUTIONS IN AN INTERDEPENDENT WORLD

Daron AcemogluJames A. Robinson

Thierry Verdier

Working Paper 18441http://www.nber.org/papers/w18441

NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts Avenue

Cambridge, MA 02138October 2012

We thank Pascual Restrepo for superb research assistance and Leopoldo Fergusson and seminar participantsat Brown University, Copenhagen University, El Rosario University in Bogotá and 2011 Latin Americanand Caribbean Economic Association meetings for comments and suggestions. Acemoglu and Robinsongratefully acknowledge support from the Canadian Institute for Advanced Research. The views expressedherein are those of the authors and do not necessarily reflect the views of the National Bureau of EconomicResearch.

NBER working papers are circulated for discussion and comment purposes. They have not been peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications.

© 2012 by Daron Acemoglu, James A. Robinson, and Thierry Verdier. All rights reserved. Short sectionsof text, not to exceed two paragraphs, may be quoted without explicit permission provided that fullcredit, including © notice, is given to the source.

Can’t We All Be More Like Scandinavians? Asymmetric Growth and Institutions in an InterdependentWorldDaron Acemoglu, James A. Robinson, and Thierry VerdierNBER Working Paper No. 18441October 2012JEL No. O33,O40,P10,P16

ABSTRACT

Because of their more limited inequality and more comprehensive social welfare systems, many perceiveaverage welfare to be higher in Scandinavian societies than in the United States. Why then does theUnited States not adopt Scandinavian-style institutions? More generally, in an interdependent world,would we expect all countries to adopt the same institutions? To provide theoretical answers to thisquestion, we develop a simple model of economic growth in a world in which all countries benefitand potentially contribute to advances in the world technology frontier. A greater gap of incomes betweensuccessful and unsuccessful entrepreneurs (thus greater inequality) increases entrepreneurial effortand hence a country’s contribution to the world technology frontier. We show that, under plausibleassumptions, the world equilibrium is asymmetric: some countries will opt for a type of “cutthroatcapitalism” that generates greater inequality and more innovation and will become the technologyleaders, while others will free- ride on the cutthroat incentives of the leaders and choose a more “cuddly”form of capitalism. Paradoxically, those with cuddly reward structures, though poorer, may have higherwelfare than cutthroat capitalists; but in the world equilibrium, it is not a best response for the cutthroatcapitalists to switch to a more cuddly form of capitalism. We also show that domestic constraints fromsocial democratic parties or unions may be beneficial for a country because they prevent cutthroatcapitalism domestically, instead inducing other countries to play this role.

Daron AcemogluDepartment of EconomicsMIT, E52-380B50 Memorial DriveCambridge, MA 02142-1347and CIFARand also NBERdaron@mit.edu

James A. RobinsonHarvard UniversityDepartment of GovernmentN309, 1737 Cambridge StreetCambridge, MA 02138and NBERjrobinson@gov.harvard.edu

Thierry VerdierPSE and ENPC 48 Boulevard Jourdan 75014 Paris Franceand CEPRverdier@pse.ens.fr

1 Introduction

Against the background of the huge inequalities across countries, the United States, Denmark,

Finland, Norway and Sweden are all prosperous, with per capita incomes more than 40 times

those of the poorest countries around the world today. Over the last 60 years, all four countries

have had similar growth rates.1 But there are also notable differences between them. The United

States is richer than Denmark, Finland and Sweden, with an income per capita (in purchasing

power parity, 2005 dollars) of about $43,000 in 2008. Denmark’s is about $35,870, Finland’s

is about $33,700 and Sweden’s stands at $34,300 (OECD, 2011).2 The United States is also

widely viewed as a more innovative economy, providing greater incentives to its entrepreneurs

and workers alike, who tend to respond to these by working longer hours, taking more risks and

playing the leading role in many of the transformative technologies of the last several decades

ranging from software and hardware to pharmaceuticals and biomedical innovations. Figure

1 shows annual average hours of work in the United States, Denmark, Finland, Norway and

Sweden since 1980, and shows the significant gap between the United States and the rest.3

Figure 1: Annual average hours worked. Source: OECD (2010)

To illustrate the differences in innovation behavior, Figure 2 plots domestic patents per

one million residents in these five countries since 1995, and shows an increasing gap between

1 In particular, the average growth rates of income per capita in the United States, Denmark, Finland, Norwayand Sweden between 1980 and 2009 are 1.59%, 1.50%, 1.94%, 2.33% and 1.56%.

2Norway, on the other hand, has higher income per capita ($48,600) than the United States, but this comparisonwould be somewhat misleading since the higher Norwegian incomes are in large part due to oil revenues.

3Average annual hours are obtained by dividing total work hours by total employment. Data from the OECDLabor market statistics (OECD, 2010).

1

the United States and the rest.4 These differences may partly reflect differential patenting

propensities rather than differences in innovativeness, or may be driven by “less important”

patents that contribute little to productive knowledge and will receive few cites (meaning that

few others will build on them). To control for this difference, we adopt another strategy.5 We

presume that important– highly-cited– innovations are more likely to be targeted to the world

market and thus patented in the US patent offi ce (USPTO). USPTO data enable us to use

citation information. Figure 3 plots the numbers of patents granted per one million residents for

Denmark, Finland, Norway and Sweden relative to the United States between 1980 and 1999.

Each number corresponds to the relevant ratio once we restrict the sample to patents that obtain

at least the number of citations (adjusted for year of grant) specified in the horizontal axis.6 If

a country is more innovative (per resident) than the United States, we would expect the gap to

close as we consider higher and higher thresholds for the number of citations. The figure shows

that, on the contrary, the gap widens, confirming the pattern indicated by Figure 2 that the

United States is more innovative (per resident) than these countries.

But there are also other important differences. The United States does not have the type of

welfare state that many European countries, including Denmark, Finland, Norway and Sweden,

have developed, and despite recent health-care reforms, many Americans do not enjoy the type

of high-quality health care that their counterparts in these other countries do. They also receive

much shorter vacations and more limited maternity leave, and do not have access to a variety of

other public services that are more broadly provided in many continental European countries.

Perhaps more importantly, poverty and inequality are much higher in the United States and have

been increasing over the last three decades, while they have been broadly stable in Denmark,

Finland, Norway and Sweden (see, e.g., Smeeding, 2002). Inequality at the top of the distribution

has also been exploding in the United States, with the top 1% of earners capturing almost 25%

of total national income, while the same number is around 5% in Finland and Sweden (Atkinson,

Piketty and Saez, 2011).

The economic and social performance of Denmark, Finland and Sweden, as well as several

other European countries, raise the possibility that the US path to economic growth is not the

4These data are from the World Intellectual Property Organization Statistics Database (WIPO 2011). TheWIPO construct these series by counting the total number of patent filings by residents in their own countrypatent offi ce. For instance, 783 patent filings per million residents in 2010 in the US is obtained by dividing thetotal number of patent filings by US citizens at the US patent offi ce (USPTO) by the number of residents inmillions. Patents are likely to be filed at different offi ces, so adding numbers from different offi ces may countthe same patent several times. Filings at own country offi ce has the advantage that it avoids multiple filings;moreover, first time filings are more likely to occur at the inventor’s home country offi ce.

5Another plausible strategy would have been to look at patent grants in some “neutral”patent offi ce or totalnumber of world patterns. However, because US innovators appear less likely to patent abroad than Europeans,perhaps reflecting the fact that they have access to a larger domestic market, this seems to create an artificialadvantage for European countries, and we do not report these results.

6Patents granted by the USPTO and citations are taken from the NBER US Patent Citations Data File.Citations are age-adjusted using the adjustment factor of Hall, Jaffe and Trajtenberg (2001), which is calculatedby estimating an obsolescence-diffusion model in which citations are explained by technology field, grant year andcitation lags. The model is then used to predict citations after the year 2006 since the data is truncated at thisdate. We do not include patents granted after 1999 so as not to excessively rely on this adjustment. For detailson the methodology and data, see Hall, Jaffe and Trajtenberg (2001) or Kerr (2008).

2

Figure 2: Patent filings per million residents at domestic offi ce. Source: World IntellectualProperty Organization.

only one, and nations can achieve prosperity within the context of much stronger safety net,

more elaborate welfare states, and more egalitarian income distributions. Many may prefer

to sacrifice 10 or 20% of GDP per capita to have better public services, a safety net, and a

more equal society, not to mention to avoid the higher pressure that the US system may be

creating.7 So can’t we all– meaning all nations of the relatively developed world– be more like

Scandinavians? Or can we?

The literature on “varieties of capitalism,”pioneered by Hall and Soskice (2001), suggests

that the answer is yes. They argue that a successful capitalist economy need not give up

on social insurance to achieve rapid growth. They draw a distinction between a Coordinated

Market Economy (CME) and a Liberal Market Economy (LME), and suggest that both have

high incomes and similar growth rates, but CMEs have more social insurance and less inequality.

Though different societies develop these different models for historical reasons and once set up

institutional complementarities make it very diffi cult to switch from one model to another, Hall

and Soskice suggest that an LME could turn itself into a CME with little loss in terms of income

and growth– and with significant gains in terms of welfare.

In this paper, we suggest that in an interconnected world, the answer may be quite different.

In particular, it may be precisely the more “cutthroat”American society that makes possible

the more “cuddly”Scandinavian societies based on a comprehensive social safety net, the wel-

fare state and more limited inequality. The basic idea we propose is simple and is developed

in the context of a canonical model of endogenous technological change at the world level. The

7Schor (1993) was among the first to point out the comparatively much greater hours that American workerswork. Blanchard (2004) has more recently argued that Americans may be working more than Europeans becausethey value leisure less.

3

Figure 3: Patents granted between 1980-1999 per million residents to each country relative tothe U.S. by number of citations. Source: NBER patent data from the USPTO.

main building block of our model is technological interdependence across countries: technolog-

ical innovations, particularly by the most technologically advanced countries, contribute to the

world technology frontier, and other countries can build on the world technology frontier.8 We

combine this with the idea that technological innovations require incentives for workers and en-

trepreneurs. From the well-known incentive-insurance trade-off captured by the standard moral

hazard models (e.g., Holmstrom, 1979), this implies greater inequality and greater poverty (and

a weaker safety net) for a society encouraging innovation. Crucially, however, in a world with

technological interdependences, when one (or a small subset) of societies is at the technological

frontier and contributing disproportionately to its advancement, the incentives for others to do

so will be weaker. In particular, innovation incentives by economies at the world technology

frontier will create higher growth by advancing the frontier, while strong innovation incentives

by followers will only increase their incomes today since the world technology frontier is already

being advanced by the economies at the frontier. This logic implies that the world equilibrium–

with endogenous technology transfer– may be asymmetric, and some countries will have greater

incentives to innovate than others. Since innovation is associated with more high-powered in-

centives, these countries will have to sacrifice insurance and equality. The followers, on the other

hand, can best respond to the technology leader’s advancement of the world technology frontier

by ensuring better insurance to their population– a better safety net, a welfare state and greater

equality.

8Such knowledge spillovers are consistent with broad patterns in the data and are often incorporated intomodels of world equilibrium growth. See, Coe and Helpman (1995) and Keller (2001), Botazzi and Peri (2003),and Griffi th, Redding and Van Reenen (2005) for some of the cross-industry evidence, and see, among others,Nelson and Phelps (1966), Howitt (2000), and Acemoglu, Aghion and Zilibotti (2006) for models incorporatinginternational spillovers.

4

The bulk of our paper formalizes these ideas using a simple (canonical) model of world equilib-

rium with technology transfer. Our model is a version of Romer’s (1990) endogenous technologi-

cal change model with multiple countries (as in Acemoglu, 2009, Chapter 18). R&D investments

within each economy advance that economy’s technology, but these build on the knowledge stock

of the world– the world technology frontier. Incorporating Gerschenkron (1962)’s famous in-

sight, countries that are further behind the world technology frontier have an “advantage of

backwardness” in that there is more unused knowledge at the frontier for them to build upon

(see also Nelson and Phelps, 1966). We depart from this framework only in one dimension: by

assuming, plausibly, that there is a moral hazard problem for workers (entrepreneurs) and for

successful innovation they need to be given incentives, which comes at the cost of consumption

insurance.9 A fully forward-looking (country-level) social planner chooses the extent of “safety

net,” which in our model corresponds to the level of consumption for unsuccessful economic

outcomes for workers (or entrepreneurs). The safety net then determines a country-level reward

structure shaping work and innovation incentives.

The main economic forces are simpler to see under two simplifying assumptions, which we

adopt in our benchmark model. First, we focus on the case in which the world technology frontier

is advanced only by the most advanced country’s technology. Second, we assume that social

planners (for each country) choose a time-invariant reward structure. Under these assumptions,

and some simple parameter restrictions (which essentially require risk aversion and the gains

from high effort to take intermediate values), we show that the world equilibrium is asymmetric:

one country (the frontier economy) adopts a “cutthroat” reward structure, with high-powered

incentives for success, while other countries free-ride on this frontier economy and choose a more

egalitarian, “cuddly,”reward structure. In the long-run, all countries grow at the same rate, but

those with cuddly reward structures are strictly poorer. Notably, however, these countries may

have higher welfare than the cutthroat leader. In fact, we prove that if the initial gap between

the frontier economy and the followers is small enough, the cuddly followers will necessarily have

higher welfare. Thus, our model confirms the intuition that all countries may want to be like

the “Scandinavians”with a more extensive safety net and a more egalitarian structure. Yet the

main implication of our theoretical analysis is that, under the assumptions of our model which

we view as a fairly natural approximation to reality, we cannot all be like the Scandinavians.

That is, it is not an equilibrium for the cutthroat leader, “the United States,” to also adopt

such a reward structure. This is because if, given the strategies of other countries, the cutthroat

leader did so, this would reduce the growth rate of the entire world economy, discouraging the

adoption of the more egalitarian reward structure. In contrast, followers are still happy to choose

more egalitarian reward structures because this choice, though making them poorer, does not

permanently reduce their growth rates, which are determined by the growth rate of the world

technology frontier shaped by innovations in the cutthroat leader.

This result makes it clear that the egalitarian reward structures in the follower countries

9To do this in the most transparent fashion, we assume that the world consists of a sequence of one-periodlived agents. We allow the social planner to have infinite horizon.

5

are made possible by the positive externalities created by the cutthroat technology leader. So

interpreting the empirical patterns in light of our theoretical framework, one may claim (with all

the usual caveats of course) that the more harmonious and egalitarian Scandinavian societies are

made possible because they are able to benefit from and free-ride on the knowledge externalities

created by the cutthroat American equilibrium.

The rest of our paper shows that our simplifying assumptions are not crucial for these main

insights, and also investigates the impact of other (domestic) institutional arrangements on the

nature of the world equilibrium. First, we characterize the equilibrium of the dynamic game

between (country-level) social planners that choose time-varying reward structures that are best

responses to the current state of the world economy and the strategies of others (more formally,

we look for the Markov perfect equilibrium of the game between the country social planners).

In this case, the equilibrium generally is time varying, but the major insights are similar. An

important difference is that in this case, we show that countries that start suffi ciently far from

the frontier will first adopt a cutthroat reward structure, and then switch to a cuddly, more

egalitarian reward structure once they approach the frontier. The reason for this is instructive.

The advantages of being backward, which are at the root of the long-run equilibrium leading

to a stable world income distribution, also imply that the return to greater innovativeness is

higher when a country is far from the world technology frontier. This encourages relatively

backward countries to also adopt a cutthroat reward structure. Nevertheless, once an economy

is suffi ciently close to the world technology frontier, the same forces as in our time-invariant

analysis kick in and encourage these follower economies to change their reward structures in

a more egalitarian, cuddly direction. Thus, under some parameter restrictions, the time path

of an economy has the flavor of the predictions of the “modernization theory,”starting with a

cutthroat reward structure and then changing in a more egalitarian direction to take advantage

of better insurance for their citizens. Nevertheless, the intuition is very different from that

of the approaches based on modernization theory, and the driving force is again the positive

externalities created by the frontier economy.10

Second, we relax the assumption that the world technology frontier is affected only by inno-

vation in the most technologically advanced country. We show that our main results extend to

this case, provided that the function aggregating the innovation decisions of all countries into

the world technology frontier is suffi ciently convex. In particular, such convexity ensures that

innovations by the more advanced countries are more important for world technological progress,

and creates the economic forces towards an asymmetric equilibrium, which is at the root of our

main result leading to an endogenous separation between cutthroat and cuddly countries.

Finally, we consider an extension in which we introduce domestic politics as a constraint

on the behavior of the social planner. We do this in a simple, reduced-form, assuming that

10 It is also worth nothing that the broad pattern implied by this analysis is in line with the fact that themore egalitarian reward structures and elements of the welfare state did not arise in follower countries integratedinto the world economy in the postwar era such as South Korea and Taiwan until they became somewhat moreprosperous.

6

in some countries there is a strong labor movement (or social democratic party) ruling out

reward structures that are very unequal. We show that if two countries start at the same level

initially, the labor movement or social democratic party in country 1 may prevent cutthroat

capitalism in that country, inducing a unique equilibrium in which country 2 is the one adopting

the cutthroat reward structure. In this case, however, this is a significant advantage, because

if the two countries start at the same level, the cutthroat country always has lower welfare.

Therefore, a tradition of strong labor movement or social democratic party, by constraining

the actions of the social planner, can act as a commitment device to egalitarianism, inducing an

equilibrium in which the country in question becomes the beneficiary from the asymmetric world

equilibrium. This result highlights that even if we cannot all be like Scandinavians, there are

certain benefits from Scandinavian-type institutions– albeit at the cost of some other country

in the world equilibrium adopting the cutthroat reward structure. This result thus also has

the flavor of the domestic political conflicts in one country being “exported”to another, as the

strength of the unions or the social democratic party in country 1 makes the poor in country 2

suffer more– as country 2 in response adopts a more cutthroat reward structure.11

It is also useful to discuss two further simplifying assumptions. First, we assume that inter-

dependence across countries is purely technological. When countries trade, those with cutthroat

incentives may specialize in different goods than those operating under cuddly reward structures.

As shown in Acemoglu and Ventura (2001), in the presence of terms of trade effects the cross-

country growth and income level implications of models of growth and trade are similar to those

with international technology spillovers (see also Acemoglu, 2009, Chapter 19), though in this

context the feedback effects between institutional choices and specialization decisions introduce

new and interesting economic forces. We leave an investigation of these issues to future research.

Second, in practice Scandinavian countries may have chosen more redistributive policies than

the US not only– not mainly– as a result of the trade-off between innovation incentives and so-

cial insurance, but also because of greater taste for redistribution or concerns of fairness among

their voters. Naturally, this does not invalidate our analysis, and such differences can be readily

incorporated into the preferences of the social planner without major changes in the formal

analysis (but, of course, the resulting equilibrium is more likely to be asymmetric). More inter-

estingly, our analysis in Section 5 shows that there will be a natural complementarity between

this type of preference for redistribution and equilibrium reward structures in a global economy.

For example, even a weak preference for redistribution might serve as a selection device, in the

same way that a strong labor movement does in Section 5, ensuring that countries with greater

preference for redistribution end up as institutional and technological followers, potentially with

positive effects on their citizens’welfare.

Our paper is related to several different literatures. First, the issues we discuss are at the core

of the “varieties of capitalism”literature in political science, e.g., Hall and Soskice (2001) which

itself builds on earlier intellectual traditions offering taxonomies of different types of capitalism

11There is a parallel between this result and Davis (1998), even though Davis takes institutions as exogenousand emphasizes a very different mechanism.

7

(Cusack, 2009) or welfare states (Esping-Anderson, 1990). A similar argument has also been

developed by Rodrik (2008). As mentioned above, Hall and Soskice (2001) argue that while both

CME and LMEs are innovative, they innovate in different ways and in different sectors. LMEs

are good at “radical innovation”characteristic of particular sectors, like software development,

biotechnology and semiconductors, while CMEs are good at “incremental innovation”in sectors

such as machine tools, consumer durables and specialized transport equipment (see Taylor, 2004,

and Akkermans, Castaldi, and Los, 2009, for assessments of the empirical evidence on these

issues). This literature has not considered that growth in an CME might critically depends on

innovation in the LMEs and on how the institutions of CMEs are influenced by this dependence.

Most importantly, to the best of our knowledge, the point that the world equilibrium may be

asymmetric, and different types of capitalism are chosen as best responses to each other, is new

and does not feature in this literature. Moreover, we conduct our analysis within the context of

a standard dynamic model of endogenous technological change and derive the world equilibrium

from the interaction between multiple countries, which is different from the more qualitative

approach of this literature.

Second, there is a related literature in economics, focusing on the causes of institutional

differences across developed economies, and on why the US lacks a European style welfare

state and why Europeans work less. Acemoglu and Pischke (1998) suggest an explanation

for differences in labor market institutions between the US and Germany based on multiple

equilibria in turnover, information and training investments. Landier (2005) develops a similar

model to account for differences in entrepreneurial risk-taking between the US and France.

Piketty (1995) proposes a model with multiple steady states, with potentially very different

redistributive policies, driven by self-fulfilling beliefs about social mobility. Bénabou and Tirole

(2006) develop a model in which self-fulfilling beliefs about justice and fairness can lead to

divergent redistributive policies across countries. Bénabou’s (2000) model is also related as it

generates multiple equilibria, one with high inequality and low redistribution and another with

low inequality and high redistribution, because redistribution can contribute to growth in the

presence of capital market imperfections (see also Saint-Paul and Verdier, 1993, and Moene and

Wallerstein, 1997). Another branch of the literature has emphasized the role of differences in

fundamentals. These include: electoral systems with proportional representation, characteristic

of continental, Europe may lead to greater redistribution (Alesina, Glaeser and Sacerdote, 2001,

Milesi-Ferretti, Perotti and Rostagno, 2002, Persson and Tabellini, 2003, Alesina and Glaeser,

2004); or the federal nature of the US may lower redistribution (Cameron, 1978, Alesina, Glaeser

and Sacerdote, 2001); or the greater ethnic heterogeneity of the US may reduce the demand

for redistribution (Alesina, Glaeser and Sacerdote, 2001, Alesina and Glaeser, 2004); or greater

social mobility in the US may mute the desire for redistributive taxation (Piketty, 1995, Bénabou

and Ok, 2001, Alesina and La Ferrara, 2005), or redistribution may be greater in Northern

Europe because of higher levels of social capital and trust (Algan, Cahuc and Sangnier, 2011).

None of this work contains the core idea of this paper that the institutions of one country interact

8

with those of another and that even with identical fundamentals asymmetric equilibria are the

norm not an exception to explain.

Third, the idea that institutional differences may emerge endogenously depending on the

distance to the world technology frontier has been emphasized in past work, for example, in

Acemoglu, Aghion and Zilibotti (2006) (see also Krueger and Kumar, 2004). Nevertheless, this

paper and others in this literature obtain this result from the domestic costs and benefits of

different types of institutions (e.g., more or less competition in the product market), and the

idea that activities leading to innovation are more important close to the world technology

frontier is imposed as an assumption. In our model, this latter feature is endogenized in a

world equilibrium, and the different institutions emerge as best responses to each other. Put

differently, the distinguishing feature of our model is that the different institutions emerge as an

asymmetric equilibrium of the world economy– while a symmetric equilibrium does not exist.

Finally, our results also have the flavor of “symmetry breaking” as in several papers with

endogenous location of economic activity (e.g., Krugman and Venables, 1996, Matsuyama, 2002,

2005) or with endogenous credit market frictions (Matsuyama, 2007). These papers share with

ours the result that similar or identical countries may end up with different choices and welfare

levels in equilibrium, but the underlying mechanism and the focus are very different.12

The rest of the paper is organized as follows. Section 2 introduces the economic environ-

ment. Section 3 presents the main results of the paper under two simplifying assumptions;

first, focusing on a specification where progress in the world technology frontier is determined

only by innovation in the technologically most advanced economy, and second, supposing that

countries have to choose time-invariant reward structures. Under these assumptions and some

plausible parameter restrictions, we show that there does not exist a symmetric world equilib-

rium, and instead, one country plays the role of the technology leader and adopts a cutthroat

reward structure, while the rest choose more egalitarian reward structures. Section 4 establishes

that relaxing these assumptions does not affect our main results. Section 5 shows how domestic

political economy constraints can be advantageous for a country because they prevent it from

adopting a cutthroat reward structure. Section 6 concludes, and proofs omitted from the text

are contained in the Appendix.

2 Model

In this section, we first describe the economic environment, which combines two components:

the first is a standard model of endogenous technological change with knowledge spillovers across

12There is also a connection between our work and the literature on “dependency theory”in sociology, developed,among others, by Cardoso and Faletto (1979) and Wallerstein (1974-2011). (We thank Leopoldo Fergusson forpointing out this connection.) This theory argues that economic development in “core” economies, such asWestern European and American ones, takes place at the expense of underdevelopment in the “periphery,” andthat these two patterns are self-reinforcing. In this theory, countries such as the United States that grow fasterare the winners from this asymmetric equilibrium. In our theory, there is also an asymmetric outcome, thoughthe mechanisms are very different, and indeed ours is more a model of “reverse dependency theory,” since it isthe Scandinavian “periphery”which, via free-riding, is benefiting from this asymmetric equilibrium.

9

J countries– and in fact closely follows Chapter 18 of Acemoglu (2009). The second introduces

moral hazard on the part of entrepreneurs, thus linking entrepreneurial innovative activity of an

economy to its reward structure. We then introduce “country social planners”who choose to

reward structures within their country in order to maximize discounted welfare.

2.1 Economic Environment

Consider an infinite-horizon economy consisting of J countries, indexed by j = 1, 2, ..., J . Each

country is inhabited by non-overlapping generations of agents who live for a period of length

∆t, work, produce, consume and then die. A continuum of agents, with measure normalized

to 1, is alive at any point in time in each country, and each generation is replaced by the next

generation of the same size. We will consider the limit economy in which ∆t → 0, represented

as a continuous time model.

The aggregate production function at time t in country j is

Yj (t) =1

1− β

(∫ Nj(t)

0xj(ν, t)

1−βdν

)Lβj , (1)

where Lj is labor input, Nj (t) denotes the number of machine varieties (or blueprints for machine

varieties) available to country j at time t. In our model, Nj (t) will be the key state variable and

will represent the “technological know-how”of country j at time t. We assume that technology

diffuses slowly and endogenously across countries as will be specified below. Finally, xj (ν, t) is

the total amount of machine variety ν used in country j at time t. To simplify the analysis,

we suppose that x depreciates fully after use, so that the x’s are not additional state variables.

Crucially, blueprints for producing these machines, captured by Nj (t), live on, and the increase

in the range of these blueprints will be the source of economic growth.

Each machine variety in economy j is owned by a technology monopolist, “entrepreneur,”

who sells machines embodying this technology at the profit-maximizing (rental) price pxj (ν, t)

within the country (there is no international trade). This monopolist can produce each unit of

the machine at a marginal cost of ψ in terms of the final good, and without any loss of generality,

we normalize ψ ≡ 1− β.Suppose that each worker/entrepreneur exerts some effort ej,i (t) ∈ {0, 1} to invent a new

machine. Effort ej,i (t) = 1 costs γ > 0 units of time, while ej,i (t) = 0 has no time cost.

Thus, entrepreneurs who exert effort consume less leisure, which is costly. We also assume

that entrepreneurial success is risky. When the entrepreneur exerts effort ej,i (t) = 1, he is

“successful” with probability q1 and unsuccessful with the complementary probability. If he

exerts effort ej,i (t) = 0, he is successful with the lower probability q0 < q1. Throughout we

assume that effort choices are private information.

We assume the utility function of entrepreneur/worker i takes the form

U (Cj,i (t) , ej,i (t)) =[Cj,i (t) (1− γej,i (t))]1−θ − 1

1− θ , (2)

10

where θ ≥ 0 is the coeffi cient of relative risk aversion (and the inverse of the intertemporal

elasticity of substitution), and this form ensures balanced growth.13

We assume that workers can simultaneously work as entrepreneurs (so that there is no

occupational choice). This implies that each individual receives wage income as well as income

from entrepreneurship, and also implies that Lj = 1 for j = 1, ..., J .

An unsuccessful entrepreneur does not generate any new ideas (blueprints), while a successful

entrepreneur in country j generates

ηN (t)φNj (t)1−φ ,

new ideas for machines, where N (t) is an index of the world technology frontier, to be endo-

genized below, and η > 0 and φ > 0 are assumed to be common across the J countries. This

form of the innovation possibilities frontier implies that the technological know-how of country

j advances as a result of the R&D and other technology-related investments of entrepreneurs in

the country, but the effectiveness of these efforts also depends on how advanced the world tech-

nology frontier is relative to this country’s technological know-how. When it is more advanced,

then the same sort of successful innovation will lead to more rapid advances, and the parameter

φ measures the extent of this.

Given the likelihood of success by entrepreneurs as a function of their effort choices and

defining ej (t) =∫ej,i (t) di, technological advance in this country can be written as:

Nj (t) = (q1ej (t) + q0(1− ej (t))) ηN (t)φNj (t)1−φ , (3)

We also assume that monopoly rights over the initial set of ideas are randomly allocated

(independent of effort) to some of the current entrepreneurs, so that they are also produced

monopolistically.14

Throughout, we maintain the following assumption:15

Assumption 1:

min{q1(1− γ)1−θ − q0 , (1− q0)− (1− q1)(1− γ)1−θ

}> 0

Finally, the world technology frontier is assumed to be given by

N (t) = G(N1 (t) , ..., NJ (t)), (4)

13When θ = 1, the utility function in (2) converges to lnCj,i (t) + ln (1− γej,i (t)). All of our results apply tothis case also, but in what follows we often do not treat this case separately to save space.14The alternative is to assume that existing machines are produced competitively. This has no impact on any

of the results in the paper, and would just change the value of B in (10) below.15This assumption ensures that, both when θ < 1 and when θ > 1, effort will only be forthcoming if entrepre-

neurs are given incentives. That is, it is suffi cient to guarantee that with the same consumption conditional onsuccess and failure, no entrepreneur would choose to exert effort. Why Assumption 1 ensures this can be seenfrom equation (9).

11

where G is a linearly homogeneous function. We will examine two special cases of this function.

The first is

G(N1 (t) , ..., NJ (t)) = max {N1 (t) , ..., NJ (t)} . (5)

which implies that the world technology frontier is given by the technology level of the most

advanced country, the technology leader, and all other countries benefit from the advances of

this technological leader. The second is a more general convex aggregator

G(N1 (t) , ..., NJ (t)) =1

J

J∑j=1

Nj (t)σ−1σ

σσ−1

, (6)

with σ < 0. As σ ↑ 0 (6) converges to (5). For much of the analysis, we focus on the simpler

specification (5), though at the end of Section 4 we show that our general results are robust

when we use (6) with σ suffi ciently small.

2.2 Reward Structures

As noted above, entrepreneurial effort levels will depend on the reward structure in each country,

which determines the relative rewards to successful entrepreneurship. In particular, suppressing

the reference to country j to simplify notation, let Rs (t) denote the time t entrepreneurial

income for successful entrepreneurs and Ru (t) for unsuccessful entrepreneurs. Thus the total

income of a worker/entrepreneur is

Ri (t) = Ri (t) + w (t) ,

where w (t) is the equilibrium wage at time t.16 In what follows, it is suffi cient to look at the total

income Ri rather than just the entrepreneurial component Ri. The reward structure can then

be summarized by the ratio r (t) ≡ Rs (t) /Ru (t). When r (t) = 1, there is perfect consumption

insurance at time t, but this generates effort e = 0. Instead, to encourage e = 1, the summary

index of the reward structure r (t) needs to be above a certain threshold, which we characterize

in the next section.

This description makes it clear that countries will have a choice between two styles of capi-

talism: “cutthroat capitalism”in which r (t) is chosen above a certain threshold, so that entre-

preneurial success is rewarded while failure is at least partly punished, and “cuddly capitalism”

in which r (t) = 1, so that there is perfect equality and consumption insurance, but this comes

at the expense of lower entrepreneurial effort and innovation.

Throughout we assume that the sequence of reward structures in country j, [rj (t)]∞t=0 is

chosen by its country-level social planner. This assumption enables us to construct a simple

game between countries (in particular, it enables us to abstract from within-country political

16Thus both Ru (t) and Rs (t) include the rents that entrepreneurs make in expectation because of existingideas being randomly allocated to them.

12

economy issues until later). Limiting the social planner to only choose the sequence of reward

structures is for simplicity and without any consequence.17

The most natural objective function for the social planner is the discounted welfare of the

citizens in that country, given by∫ ∞0

e−ρt(∫

U (Cj,i (t) , ej,i (t)) di

)dt =

∫ ∞0

e−ρt

(∫[Cj,i (t) (1− γej,i (t))]1−θ − 1

1− θ di

)dt,

(7)

where ρ is the discount rate that the social planner applies to future generations and

U (Cj,i (t) , ej,i (t)) denotes the utility of agent i in country j alive at time t (and thus the

inner integral averages across all individuals of that generation). One disadvantage of this ob-

jective function is that it imposes that the inverse of the intertemporal elasticity of substitution

is equal to the coeffi cient of relative risk aversion. Since comparative statics with respect to the

coeffi cient of relative risk aversion are of interest, this feature is not desirable. For this reason,

we will also discuss the results with Epstein-Zin preferences for social planners (Epstein and Zin,

1989), which separate the coeffi cient of relative risk aversion from the intertemporal elasticity

of substitution. These preferences, in continuous time, take the form:∫ ∞0

e−ρt

[∫ ([Cj,i (t) (1− γej,i (t))]1−θ

1− θ

)di

] 1−λ1−θ

dt

1

1−λ

, (8)

where θ is still the coeffi cient of relative risk aversion, but now the inverse of the intertemporal

elasticity of substitution is given by λ. When λ = θ, we got back to (7).

3 Equilibrium with Time-Invariant Reward Structures

In this section, we simplify the analysis by assuming that the reward structure for each country j

is time-invariant, i.e., rj (t) = rj , and is chosen at time t = 0. This assumption implies that each

country chooses between “cuddly”and “cutthroat”capitalism once and for all, and enables us

to characterize the structure of the world equilibrium in a transparent manner, showing how this

equilibrium often involves different choices of reward structures across countries– in particular,

one country choosing cutthroat capitalism while the rest choose cuddly capitalism. In this

section, we focus on the “max”specification of the world technology frontier given by (5), and

we also start with the simpler objective for the social planner given by the discounted utilities

of different generations as in (7).

3.1 World Equilibrium Given Reward Structures

We first characterize the dynamics of growth for given (time-invariant) reward structures. The

following proposition shows that a well-defined world equilibrium exists and involves all coun-

tries growing at the same rate, set by the rate of growth of the world technology frontier. This17 If we allow the social planner to set prices that prevent the monopoly markup, nothing in our analysis below,

except that the value of B in (10), would change.

13

growth rate is determined by the innovation rates (and thus reward structures) of either all coun-

tries (with (6)) or the leading country (with (5)). In addition, differences in reward structures

determine the relative income of each country.

Proposition 1 Suppose that the reward structure for each country is constant over time (i.e.,for each j, Rjs (t) /Rju (t) = rj). Then starting from any initial condition (N1 (0) , ..., NJ (0)),

the world economy converges to a unique stationary distribution (n∗1, ..., n∗J), where nj (t) ≡

Nj (t) /N (t) and N (t) /N (t) = g∗, and (n∗1, ..., n∗J) and g∗ are functions of (r1, ..., rJ). Moreover,

with the max specification of the world technology frontier, (5), g∗ is only a function of the most

innovative country’s reward structure, r`.

Proof. The proof of this proposition follows from the material in Chapter 18 of Acemoglu

(2009) with minor modifications and is omitted to save space.

The process of technology diffusion ensures that all countries grow at the same rate, even

though they may choose different reward structures. In particular, countries that do not en-

courage innovation will first fall behind, but given the form of technology diffusion in equation

(3), the advances in the world technology frontier will also pull them to the same growth rate

as those that provide greater inducements to innovation. The proposition also shows that in

the special case where (5) applies, it will be only innovation and the reward structure in the

technologically most advanced country that determines the world growth rate, g∗.

3.2 Cutthroat and Cuddly Reward Structures

We now define the cutthroat and cuddly reward structures. Consider the reward structures that

ensure effort e = 1 at time t. This will require that the incentive compatibility constraint for

entrepreneurs be satisfied at t, or in other words, expected utility from exerting effort e = 1

should be greater than expected utility from e = 0. Using (2), this requires

1

1− θ

(q1Rs (t)1−θ + (1− q1)Ru (t)1−θ

)(1− γ)1−θ ≥ 1

1− θ

(q0Rs (t)1−θ + (1− q0)Ru (t)1−θ

),

where recall that Rs (t) is the income and thus the consumption of an entrepreneur/worker

conditional on successful innovation, and Ru (t) is the income level when unsuccessful, and this

expression takes into account that high effort leads to success with probability q1 and low effort

with probability q0, but with high effort the total amount of leisure is only 1− γ. Rearrangingthis expression, we obtain

r (t) ≡ Rs (t)

Ru (t)≥

((1− q0)− (1− q1)(1− γ)1−θ

q1(1− γ)1−θ − q0

) 11−θ

=

(1 +

1− (1− γ)1−θ

q1(1− γ)1−θ − q0

) 11−θ

≡ A. (9)

Clearly, the expression A defined in (9) measures how “high-powered”the reward structure needs

to be in order to induce effort, and will thus play an important role in what follows. Assumption

14

1 is suffi cient to ensure that A > 1.18

Since the social planner maximizes average utility, she would like to achieve as much con-

sumption insurance as possible subject to the incentive compatibility constraint (9), which im-

plies that she will satisfy this constraint as equality. In addition, Rs (t) and Ru (t) must satisfy

the resource constraint at time t. Using the expression for total output and expenditure on

machines provided in the Appendix, this implies

q1Rs (t) + (1− q1)Ru (t) = BNj (t)

where

B ≡ β(2− β)

1− β , (10)

and we are using the fact that in this case, all entrepreneurs will exert high effort, so a fraction

q1 of them will be successful. Combining this expression with (9), we obtain

Rs (t) =BA

q1A+ (1− q1)Nj (t) and Ru (t) =

B

q1A+ (1− q1)Nj (t) . (11)

The alternative to a reward structure that encourages effort is one that forgoes effort and

provides full consumption insurance– i.e., the same level of income to all entrepreneur/workers

of R0 (t), regardless of whether they are successful or not. In this case, the same resource

constraint implies

R0 (t) = BNj (t) . (12)

Given these expressions, the expected utility of entrepreneurs/workers under the “cutthroat”

and “cuddly”capitalist systems, denoted respectively by s = c and s = o, can be rewritten as

W cj (t) ≡ E

[U(Ccj (t) , ecj (t)

)]=

(q1Rs (t)1−θ + (1− q1)Ru (t)1−θ

)(1− γ)1−θ − 1

1− θ ,

W oj (t) ≡ E

[U(Coj (t) , eoj (t)

)]=R0 (t)1−θ − 1

1− θ .

Now using (11) and (12), we can express these expected utilities as:19

W cj (t) = ωcNj (t)1−θ − 1

1− θ and Woj (t) = ωoNj (t)1−θ − 1

1− θ ,

where

ωc ≡(q1A

1−θ + (1− q1))

(1− γ)1−θ

(q1A+ (1− q1))1−θB1−θ

1− θ and ωo ≡B1−θ

1− θ . (13)

It can be verified that ωc < ωo, though when θ > 1, it important to observe that we have

ωc < ωo < 0. It can also be established straightforwardly that ωc, and thus (ωc/ωo)1/(1−θ), is

decreasing in A, defined in (9), (since a higher A translates into greater consumption variability);

18 In particular, when θ < 1, 1+ 1−(1−γ)1−θq1(1−γ)1−θ−q0

is greater than one and is raised to a positive power, while whenθ > 1, it is less than one and it is raised to a negative power.19 In what follows, we will also drop the constant −1/ (1− θ) in W c

j (t) and Woj (t) when this causes no confusion.

15

that A (and thus (ωc/ωo)1/(1−θ)) is increasing in γ (to compensate for the higher cost of effort);

and that A is non-monotone in θ (because a higher coeffi cient of relative risk aversion also

reduces the disutility of effort).

From (3), the growth rate of technology of country j adopting reward structure sj ∈ {c, o}can be derived as

Nj (t) = gsjN (t)φNj (t)1−φ

where the growth rates gsj ∈ {gc, go} are given by

go ≡ q0η, and gc ≡ q1η.

This reiterates that at any point in time, a country choosing a cutthroat reward structure will

have a faster growth of its technology stock.

We next introduce a second assumption, which ensures that the cutthroat growth rate, gc,

is not so high as to lead to infinite welfare for the country social planners and will also be

maintained throughout (without explicitly being stated):

Assumption 2:ρ− (1− θ) gc > 0.

3.3 Equilibrium Reward Structures

We now characterize the equilibrium of the game between the country social planners. Since re-

ward structures are chosen once and for all at time t = 0, the interactions between the country so-

cial planners can be represented as a static game with the payoffs given as the discounted payoffs

implied by the reward structures of all countries (given initial conditions {N1 (0) , ..., NJ (0)}).We will characterize the Nash equilibria of this static game. We also restrict attention to the sit-

uation in which the same country, denoted `, remains the technology leader throughout. Given

our focus on the world technology frontier specification in (5), the fact that this country is the

leader implies at each t implies that N` (t) = max {N1 (t) , ..., NJ (t)} for all t. This assumptionsimplifies the exposition in this section.20

We next introduce the key condition which will ensure that the technology leader prefers

a cutthroat reward structure. It is straightforward to verify that when this condition is not

satisfied, all countries will choose a cuddly reward structure. Thus this condition restricts

attention to the interesting part of the parameter space.

Condition 1:ωc

ρ− (1− θ)gc>

ωoρ− (1− θ)go

.

20Essentially, it enables us to pick a unique equilibrium among asymmetric equilibria. A byproduct of theanalysis in Section 5 is to show how this assumption can be relaxed without affecting our main results.

16

Why this condition ensures that the technology leader, country `, prefers a cutthroat reward

structure can be seen straightforwardly by noting that when the growth rate of the world tech-

nology frontier is determined by innovation in country `, ωcρ−(1−θ)gc is the discounted value from

such a cutthroat reward structure, while the discounted value of a cuddly reward structure isωo

ρ−(1−θ)go given that all other countries are choosing a cuddly strategy.21

Now recalling that nj(t) ≡ Nj (t) /N (t) = Nj (t) /N` (t), for j 6= ` we have

nj(t)

nj(t)=

(N` (t)

Nj (t)

)φgsj − g` = nj(t)

−φgsj − g`.

where g` = gc, and we have imposed that the leader is choosing a cutthroat reward structure.

This differential equation’s solution is

Nj (t) =

(Nj (0)φ +

gsjgc

(eφgct − 1

)(N` (0))φ

) 1φ

, (14)

enabling us to evaluate the welfare of the country j social planner from reward structure sj ∈{c, o}:

Wj(sj) =

∫ ∞0

e−ρtWsjj (t) =

∫ ∞0

e−ρtωsjN` (0)1−θ(nj(0)φ +

gsjgc

(eφgct − 1

)) 1−θφ

dt (15)

= ωsjN` (0)1−θ(gsjgc

) 1−θφ∫ ∞

0e−(ρ−(1−θ)gc)t

(1 +

(gcgsj

nj (0)φ − 1

)e−φgct

) 1−θφ

dt ,

where recall that nj(0) ≡ Nj (0) /N` (0). The second line of (15) highlights that, under Condition

1, the long-run growth rate of all countries will be gc, and thus ensure that these welfare levels

are well defined. This implies the following straightforward result:

Proposition 2 Suppose that each country chooses a time-invariant reward structure at timet = 0. Suppose also that country social planners maximize (7), the world technology frontier is

given by (5), Condition 1 holds, and

(ωcωo

) 11−θ

<

(gogc

) 1φ

∫∞

0 e−(ρ−(1−θ)gc)t(

1 +(gcgonj (0)φ − 1

)e−φgct

) 1−θφdt∫∞

0 e−(ρ−(1−θ)gc)t(

1 +(nj (0)φ − 1

)e−φgct

) 1−θφdt

1

1−θ

for each j 6= `.

(16)

Then there exists no symmetric equilibrium. Moreover, there exists a unique world equilibrium in

which the initial technology leader, country ` remains so throughout, and this equilibrium involves

country ` choosing a cutthroat reward structure, while all other countries choose a cuddly reward

structure. In this world equilibrium, country ` grows at the rate gc throughout, while all other

countries asymptotically grow also at this rate, and converge to a level of income equal to a

fraction go/gc of the level of income of country `.21 If the country in question chose a cuddly reward structure while some other country chose the cutthroat

structure, then this other country would necessarily become the leader at some point. Here we are restrictingattention to the case in which this other country would be the leader from the beginning, which is without muchloss of generality.

17

Proof. Suppose first that country ` chooses a cutthroat reward structures throughout. Thenthe result that country j strictly prefers to choose a cuddly reward structure follows immediately

from comparing Wj(c) and Wj(o) given by (15) (remembering that when θ > 1, we have ωc <

ωo < 0 and thus the direction of inequality is reversed twice, first when we divide by ωo and

second when we raise the left-hand side to the power 1/ (1− θ)).The result that there exists no symmetric equilibrium in which all countries choose the same

reward structure follows from this observation: when (16) holds, all j 6= ` will choose a cuddly

reward structure when ` chooses a cutthroat one; and Condition 1 implies that when j 6= `

choose a cuddly reward structure, country ` strictly prefers to choose a cutthroat one. This also

characterizes the unique equilibrium in which ` remains the technology leader throughout.

Finally, convergence to a unique stationary distribution of income with the same asymptotic

growth rate follows from Proposition 1, and the ratio of income between the leader and followers

in this stationary distribution is given from the limit of equation (14).

The important implication is that, under the hypotheses of the proposition, the world equilib-

rium is necessarily asymmetric– i.e., a symmetric equilibrium does not exist. Rather, it involves

one country choosing a cutthroat reward structure, while all others choose cuddly reward struc-

tures. The intuition for this result comes from the differential impacts of the leader, country `,

and non-leader countries on the world growth rate. Because country `’s innovations and reward

structure determine the pace of change of the world technology frontier, if it were to switch from

a cutthroat to a cuddly reward structure, this would have a growth effect on the world economy

(and thus on itself). The prospect of permanently lower growth discourages country ` from

choosing a cuddly reward structure. In contrast, any other country deviating from the asym-

metric equilibrium and choosing a cutthroat reward structure would only enjoy a beneficial level

effect : such a country would increase its position relative to country `, but would not change

its long-run growth rate (because its growth rate is already high thanks to the spillovers from

the cutthroat incentives that country ` provides to its entrepreneurs). The contrast between the

growth effect of the reward structure of the leader and the level effect of the reward structure

of followers is at the root of the asymmetric equilibrium (and the non-existence of asymmetric

equilibrium).22

Condition (16), which ensures that the world equilibrium is asymmetric, is in terms of the

ratio of two integrals which do not in general have closed-form solutions. Nevertheless, the

special case where φ = 1 − θ admits a closed-form solution and is useful to illustrate the main

22Naturally, with any asymmetric equilibrium of this type, there are in principle several equilibria, with onecountry playing the role of the leader and choosing a cutthroat reward structure, while others choose cuddlyreward structures. Uniqueness here results from the fact that we have imposed that the same country remainsthe leader throughout, which picks the initial technology leader as the country choosing the cutthroat rewardstructure.

18

insights. In particular, in this case (16) simplifies to :

(ωcωo

)<

(gogc

)∫∞0 e−(ρ−φgc)t(

1 +(gcgonj (0)φ − 1

)e−φgct

)dt∫∞

0 e−(ρ−φgc)t(

1 +(nj (0)φ − 1

)e−φgct

)dt

=nj (0)φ (ρ− φgc) + φgo

nj (0)φ (ρ− φgc) + φgc

(17)

Inspection of (17) shows that an asymmetric equilibrium is more likely to emerge when nj (0) is

close to 1 for all followers– since the last expression is strictly increasing in nj (0). This implies

that, bearing in mind that country ` is the technology leader initially, the asymmetric equilibrium

is more likely to emerge when all countries are relatively equal to start with. Intuitively, the

innovation possibilities frontier (3) implies that a country that is further behind the world

technology frontier (i.e., low nj (0)) has a greater growth potential– and in fact will grow faster

for a given level of innovative activity. This also implies that the additional gain in growth

from choosing a cutthroat reward structure is greater the lower is nj (0). Consequently, for

countries that are significantly behind the world technology frontier (or behind country `), the

incentives to also adopt a cutthroat reward structure are stronger.23 The next figure illustrates

this diagrammatically. The upward sloping curve plots the right-hand side of (17) as a function

of nj (0). When this expression is equal to ωc/ωo, country j is indifferent between a cuddly and

cutthroat reward structure, and for countries technologically more advanced than the threshold

level n, a cuddly reward structure is preferred and the asymmetric equilibrium emerges.

Figure 4: Choice of cutthroat and cuddly reward structures as a function of technology gapnj (0).

We next provide a simpler suffi cient condition that enables us to reach the same conclusion

as in Proposition 2.

23We will see in the next section that this economic force will sometimes lead to a time-varying reward structure.

19

Corollary 1 1. The condition (ωcωo

) 11−θ

<

(gogc

) 1φ

(18)

is suffi cient for (16) to hold, so under this condition and the remaining hypotheses of

Proposition 1, the conclusions in Proposition 1 hold.

2. In addition, there exists n such that for nj (0) < n, the condition(ωcωo

) 11−θ

<

(gogc

) 1φ[1 +

nj(0)φ

φ(ρ− (1− θ)gc)

(1

go− 1

gc

)] 11−θ

(19)

is suffi cient for (16) to hold, so under this condition and the remaining hypotheses of

Proposition 1, the conclusions in Proposition 1 hold.

Proof. We first proved the second part of the corollary. By integration by parts,∫ ∞0

e−ρt(eφgct − 1

) 1−θφdt =

[−e−ρt

ρ

(eφgct − 1

) 1−θφ

]∞0

+

∫ ∞0

e−ρt

ρ(1− θ) eφgctgc

(eφgct − 1

) 1−θφ−1dt

=

∫∞0 e−ρt

(eφgct − 1

) 1−θφ−1dt

ρ−(1−θ)gc(1−θ)gc

.

Then, a first-order Taylor approximation of (15) for nj(0) ≡ Nj (0) /Nl (0) small gives

Wj (sj) =

∫ ∞0

e−ρtωsjN` (0))1−θ[(gsjgc

(eφgct − 1

))1−θφ +

1− θφ

nj(0)φ(gsjgc

(eφgct − 1

))1−θφ−1]dt+R (nj (0))

= ωsj (N` (0)1−θ(gsjgc

) 1−θφ(

1 +nj(0)φ

φ

ρ− (1− θ) gcgsj

)∫ ∞0

e−ρt(eφgct − 1

) 1−θφdt+R (nj (0)) ,

where R (nj (0)) is the residual which goes to zero as nj (0)→ 0. Thus there exists n > 0 such

that for nj (0) < n, Wj (c) <Wj (o) if

(ωcωo

) 11−θ

<

(gogc

) 1φ

1 +nj(0)φ

φρ−(1−θ)gc

go

1 +nj(0)φ

φρ−(1−θ)gc

gc

11−θ

.

Next another first-order Taylor approximation of the right-hand side of this expression gives

(18), and with the same reasoning implies that there exists some 0 < n ≤ n such that for

nj (0) < n, Wj (c) <Wj (o) if (18) holds.

The second part now follows by setting nj (0) = 0, and noting that in the right-hand side of

(16), the term in large parentheses (the ratio of the two integrals) is always greater than 1, so

that(ωcωo

) 11−θ

<(gogc

) 1φis a suffi cient condition for all nj (0).

We next provide a simple result characterizing when Condition 1 (which ensures that the

leader prefers a cutthroat reward structure) and (18) (which ensures that followers choose a

cuddly reward structure) are simultaneously satisfied. This result illustrates the role of risk

aversion in the asymmetric equilibria described above.24

24Recall, however, that (18) is a suffi cient condition– not the exact condition– for such a symmetric equilibriato exist.

20

Corollary 2 1. Condition (18) is satisfied for γ ∈ (0, γ) (where γ ≡ 1 −√

q0(1−q0)q1(1−q1)) and

θ ≥ θ∗(φ, γ) where γ > 0 and 0 < θ∗(φ, γ) <∞. Moreover θ∗(φ, γ) is decreasing in φ and

γ.

2. Condition 1 is satisfied for θ ∈[0, θ]and ρ ∈ (ρ(θ, γ), (1 − θ)gc), where θ > 0. Moreover

ρ(θ, γ) is decreasing in θ and in γ.

Proof. See the Appendix.Therefore, this corollary implies that, under the specified conditions, the asymmetric equi-

librium will arise (or more accurately, the suffi cient conditions for an asymmetric equilibria will

be satisfied) when θ ≥ θ∗(φ, γ), i.e., when the coeffi cient of relative risk aversion is suffi ciently

high. But to ensure Assumption 2 also holds, this coeffi cient needs to be less than some thresh-

old θ > 1. Note, however, that as φ increases (so that there are greater technology spillovers

from the leader to followers), θ∗(φ, γ) decreases, making these conditions more likely to be sat-

isfied. Naturally, as the second part of the corollary specifies, we also need ρ not to be too

small, otherwise it would not be a best response for the technology leader to choose a cutthroat

reward structure. Though intuitive, this corollary suffers from the feature that changes in θ not

only correspond to changes in the coeffi cient of relative risk aversion but also to changes in the

intertemporal elasticity of substitution. For this reason, the next subsection considers the more

general preferences in (8) which separate these two parameters.

Remark 1 We have so far restricted countries to choose either cutthroat or cuddly reward

structures for all of their entrepreneurs. In the next section, we allow for mixed reward structures

whereby some entrepreneurs are given incentives to exert high effort, while others are not. It is

straightforward to see that in this case, (16) continues to be suffi cient, together with Assumptions

1-3, for there not to exist a symmetric equilibrium, but is no longer necessary. Suffi ciency

follows simply from the following observation: condition (16) implies that for followers a cuddly

reward structure is preferred to a cutthroat one, so even when intermediate reward structures

are possible, the equilibrium will not involve a cutthroat reward structure, hence will not be

symmetric. The reason why (16) is not necessary is that when φ > 1− θ, welfare is concave inthe fraction of agents receiving cutthroat incentives (as we show in the next section), and thus

even if a cuddly reward structure is not preferred to a cutthroat one, an intermediate one may

be. In particular, denoting the fraction of entrepreneurs receiving cutthroat incentives by u, the

necessary condition for the follower to adopt a cuddly reward structure is

∂Wj (u = 0)

∂u= (ωc − ωo)

∫ ∞0

e−ρt(nj(0)φ + eφgct − 1

) 1−θφdt

+(1− θ)ωc(gc − go)

φgc

∫ ∞0

e−ρt(nj(0)φ +

(eφgct − 1

)) 1−θφ−1 (

eφgct − 1)dt < 0.

We can also note that under Assumptions 1 and 2 and Condition 1, there cannot be a fully

mixed reward structure equilibrium where all countries choose a fraction u∗ of entrepreneurs to

21

receive cutthroat incentives. Suppose that all countries, except the technology leader, choose a

mixed reward structure with the fraction u∗ of entrepreneurs receiving cutthroat incentives. If

the leader also chose u∗, it would remain the leader forever, with discounted utility of

W` (u∗) =ωcu

∗ + ωo(1− u∗)ρ− (1− θ) (gcu∗ + go(1− u∗))

which is strictly increasing in u∗, so that the leader would in fact prefer fully cutthroat rewards.

3.4 The Effects of Risk Aversion

To study the effects of risk aversion, we first show that the results derived so far apply with the

Epstein-Zin preferences for this social planner as in (8) (Epstein and Zin, 1989). To do this, we

first need to modify Assumption 2 and Condition 1 by substituting for λ instead of θ (since it

is the intertemporal elasticity of substitution that matters in this case):

Assumption 2′:

ρ− (1− λ) gc > 0.

Condition 1′: (

ωcωo

) 11−θ

>

[ρ− (1− λ)gcρ− (1− λ)go

] 11−λ

.

We can then establish a generalization of Proposition 2 that applies with social planner

preferences given by (8).

Proposition 3 Suppose that each country chooses a time-invariant reward structure at timet = 0. Suppose also that country social planners maximize (8); the world technology frontier is

given by (5); Assumptions 1 and 2′and Condition 1

′hold; and

(ωcωo

) 11−θ

<

(gogc

) 1φ

∫∞

0 e−(ρ−(1−λ)gc)t(

1 +(gcgonj (0)φ − 1

)e−φgct

) 1−λφdt∫∞

0 e−(ρ−(1−λ)gc)t(

1 +(nj (0)φ − 1

)e−φgct

) 1−λφdt

1

1−λ

for each j 6= `.

(20)

Then there exists no symmetric equilibrium. Moreover, there exists a unique world equilibrium in

which the initial technology leader, country ` remains so throughout, and this equilibrium involves

country ` choosing a cutthroat reward structure, while all other countries choose a cuddly reward

structure. In this world equilibrium, country ` grows at the rate gc throughout, while all other

countries asymptotically grow also at this rate, and converge to a level of income equal to a

fraction go/gc of the level of income of country `.

22

Proof. The proof is essentially identical to that of Proposition 2 after noting that with (8),following the same steps as above, welfare is given by:

Wj(sj) =

[∫ ∞0

e−ρt(W

sjj (t)

) 1−λ1−θ

dt

] 11−λ

= ω1

1−θsj N` (0)

(gsjgc

) 1φ

[∫ ∞0

e−(ρ−(1−λ)gc)t

(1 +

(gcgsj

nj (0)φ − 1

)e−φgct

) 1−λφ

dt

] 11−λ

.

Therefore, with the more general Epstein-Zin preferences given in (8), Proposition 2 remains

essentially unchanged except that λ, as the inverse of the intertemporal elasticity of substitution,

replaces θ in the integrals. This also immediately implies that Corollary 1 also applies in this

case and we do not state it to save space. The more interesting implication of these more general

preferences arises when we turn to the implications of changes in the coeffi cient of relative risk

aversion. The following corollary strengthens Corollary 2 and shows that asymmetric equilibria

are more likely to arise for intermediate values of the coeffi cient of relative risk aversion.

Corollary 3 Suppose γ ≤ γ ≡ 1−√

q0(1−q0)q1(1−q1) . Then Conditions 1

′and (20) are jointly satisfied

for intermediate values of the coeffi cient of relative risk aversion θ.

Proof. Conditions 1′and (20) are jointly satisfied if

(ρ− (1− λ)gcρ− (1− λ)go

) 11−λ

<

(ωcωo

) 11−θ

<

(gogc

) 1φ

∫∞

0 e−(ρ−(1−λ)gc)t(

1 +(gcgonj (0)φ − 1

)e−φgct

) 1−λφdt∫∞

0 e−(ρ−(1−λ)gc)t(

1 +(nj (0)φ − 1

)e−φgct

) 1−λφdt

1

1−λ

.

Only the middle term depends on θ. Straightforwardly, this term is decreasing in θ for given A

and also decreasing in A. Moreover, as shown in the proof of Corollary 2, A is decreasing in θ for

γ ≤ γ. Thus(ωcωo

) 11−θ

is monotonically decreasing in θ. Furthermore, limθ→θmax

(ωcωo

) 11−θ

= 0

for θmax ≡ 1−log(

1−q01−q1

)

log(1−γ) > 1, establishing the corollary.

This result is intuitive: if the coeffi cient of relative risk aversion is too high, then no country,

not even those at the frontier, would adopt cutthroat incentives, and if there is very limited risk

aversion, then all countries are more likely to choose cutthroat incentives.

To gain further intuition, we can again plot the equivalent of the right-hand side of (17), but

now under the restriction that φ = 1 − λ, which leaves the coeffi cient of relative risk aversion,θ, free. In particular, the equivalent of (17) is now:(

ωcωo

) 11−θ

<

((ρ− φgc)nj (0)φ + φgo

(ρ− φgc)nj (0)φ + φgc

) 1φ

. (21)

The next figure plots the left and right-hand sides of (21) and also illustrates that now an

increase in risk aversion (an increase in θ) shifts the left-hand side up (again presuming that

γ ≤ γ), expanding the set of technology gaps at which the asymmetric equilibrium arises.

23

Figure 5: The effect of risk aversion on equilibrium reward structure with Epstein-Zin prefer-ences.

3.5 Welfare

The most interesting result concerning welfare is that, even though the technological leader,

country `, starts out ahead of others and chooses a “growth-maximizing” strategy, average

welfare (using the social planner’s discount rate) may be lower in that country than in the

followers choosing a cuddly reward structure. This result is contained in the next proposition and

its intuition captures the central economic force of our model: followers are both able to choose

an egalitarian reward structure providing perfect insurance to their entrepreneur/workers and

benefit from the rapid growth of technology driven by the technology leader, country `, because

they are able to free-ride on the cutthroat reward structure in country `, which is advancing the

world technology frontier. In contrast, country `, as the technology leader, must bear the cost

of high risk for its entrepreneur/workers. The fact that followers prefer to choose the cuddly

reward structure implies that, all else equal, the leader, country `, would have also liked to

but cannot do so, because it realizes that if it did, the growth rate of world technology frontier

would slow down– while followers know that the world technology frontier is being advanced by

country ` and can thus free-ride on that country’s cutthroat reward structure.

Proposition 4 Suppose that countries are restricted to time-invariant reward structures, andCondition 1 and (16) (or Condition 1

′and (20)) hold, so that country ` adopts the cutthroat

strategy and country j adopts the cuddly strategy. Then their exists δ > 0 such that for all

nj (0) > 1− δ, welfare in country j is higher than welfare in country `.

Proof. Consider the case where n` (0) = nj (0). Then the result follows immediately from (16)

(or (20)), since, given this condition, country j strictly prefers to choose a cuddly rather than a

24

cutthroat reward structure. If it were to choose a cutthroat structure, it would have exactly the

same welfare as country `. Next by continuity, this is also true for nj (0) > 1−δ for δ suffi cientlysmall and positive.

4 Equilibrium with Time-Varying Rewards Structures

In this section, we relax the assumption that reward structures are time-invariant, and thus

assume that each country chooses sj(t) ∈ {c, o} at time t, given the strategies of other countries,thus defining a differential game among the J countries. We focus on the Markov perfect

equilibria of this differential game, where strategies at time t are only conditioned on payoff

relevant variables, given by the vector of technology levels. To start with, we focus on the world

technology frontier given by (5), and at the end, we will show that the most important insights

generalize to the case with general aggregators of the form (6) provided that these aggregators

are suffi ciently “convex,”i.e., putting more weight on technologically more advanced countries.

Throughout this section, we also suppose that country social planners maximize (7).

4.1 Main Result

In this subsection, we focus on the world technology frontier given by (5), and also assume that

at the initial date, there exists a single country ` that is the technology leader, i.e., a single ` for

which N` (0) = max {N1 (0) , ..., NJ (0)}. We also allow follower countries to provide cutthroatreward structures to some of their entrepreneurs while choosing a cuddly reward structure for

the rest. Hence, we define uj (t) as the fraction of entrepreneurs receiving a cutthroat reward

structure,25 and thus

ω (uj(t)) = ωo(1− uj(t)) + ωcuj(t)

g((uj(t)) = go(1− uj(t)) + gcuj(t),

with uj (t) ∈ [0, 1], and uj (t) = 0 at all points in time corresponds to a cuddly reward structure

and uj (t) = 1 for all time is cutthroat throughout, like those analyzed in the previous section.

The problem of the country j social planner can then be written as

Wj(Nj (t) ,N` (t)) = maxuj(·)∈[0,1]

∫ ∞t

e−ρ(τ−t)ω (uj(τ))Nj (τ)1−θ dτ (22)

such that Nj (τ) = g((uj(τ)) N` (τ)φNj (τ)1−φ ,

with N` (τ) = N (t) egc(τ−t) (for τ ≥ t).

Depending on what the country j social planner can condition on for the choice of time t reward

structure, this would correspond to either a “closed loop” or “open loop”problem– i.e., one

in which the strategies are chosen at the beginning or are updated as time goes by. In the

25 It is straightforward to see that it is never optimal to give any entrepreneur any other reward structures thanperfect insurance or the cutthroat reward structure that satisfies the incentive compatibility constraint as equality

25

Appendix, we show that the two problems have the same solution, so the distinction is not

central in this case. The main result in this section is as follows.

Proposition 5 Suppose country social planners maximize (7), the world technology frontier isgiven by (5), Condition 1 holds, and technology spillovers are large in the sense that φ > 1− θ.Let

m ≡ (1− θ)(ωo − ωc) gc + (gc − go)ωc(ωo − ωc) (ρ+ φgc)

. (23)

Then the world equilibrium is characterized as follows:

1. If

m <gogc, (24)

there exist m < go/gc and 0 < T < ∞ such that for nj (0) < m1/φ, the reward structure

of country j is cutthroat (i.e., sj(t) = c or uj (t) = 1) for all t ≤ T , and cuddly (i.e.,

sj(t) = o or uj (t) = 0) for all t > T ; for nj (0) ≥ m1/φ, the reward structure of country

j is cuddly (i.e., sj(t) = o or uj (t) = 0) for all t. Moreover, m > 0 if θ < 1, and m < 0

if θ is suffi ciently large (in which case the cuddly reward structure applies with any initial

condition). Regardless of the initial condition (and the exact value of m), in this case,

nj (t)→ (go/gc)1/φ.

2. Ifgogc< m < 1, (25)

there exists 0 < T < ∞ such that for nj (0) < m1/φ, the reward structure of country

j is cutthroat (i.e., sj(t) = c or uj (t) = 1) for all t ≤ T , and then at t = T when

nj (T ) = m1/φ, the country adopts a “mixed” reward structure and stays at nj (t) = m1/φ

(i.e., uj (t) = u∗j ∈ (0, 1)) for all t > T ; for nj (0) > m1/φ, the reward structure of

country j is cuddly (i.e., sj(t) = o or uj (t) = 0) for all t ≤ T , and then at t = T when

nj (T ) = m1/φ, the country adopts a mixed reward structure and stays at nj (t) = m1/φ

(i.e., uj (t) = u∗j ∈ (0, 1)) for all t > T .

3. If

m > 1, (26)

then the reward structure of country j is cutthroat for all t (i.e., sj(t) = c or uj (t) = 1 for

all t).

Proof. We prove the more general Proposition 6 in the next section. A different and more

direct proof of Proposition 5 is provided in the Appendix.

This proposition has several important implications. First, the equilibrium of the previous

section emerges as a special case, in particular when condition (24) holds and the initial gap

between the leader and the followers is not too large (i.e., nj (0) is greater than the threshold

26

specified in the proposition), or whenm < 0. In this case, the restriction to time-invariant reward

structures is not binding, and exactly the same insights as in the previous section obtain.

Secondly, however, the rest of the proposition shows that the restriction to time-invariant

reward structures is generally binding, and the equilibrium involves countries changing their

reward structures over time. In fact, part 1 of the proposition shows that, in line with the

discussion following Proposition 2, the growth benefits of cutthroat reward structures are greater

when the initial gap between the leader and the country in question is larger, because this creates

a period during which this country can converge rapidly to the level of income of the technological

leader, and a cutthroat reward structure can significantly increase this convergence growth rate.

In consequence, for a range of parameters, the equilibrium involves countries that are suffi ciently

behind the technological leader choosing cutthroat reward structures, and then after a certain

amount of convergence takes place, switching to cuddly capitalism. This pattern, at least from

a bird’s eye perspective, captures the sort of growth and social trajectory followed by countries

such as South Korea and Taiwan, which adopted fairly high-powered incentives with little safety

net during their early phases of convergence, but then started building a welfare state.

Thirdly, part 2 shows that without the restriction to time-invariant reward structures, some

countries may adopt mixed reward structures when they are close to the income level of the

leader. With such reward structures some entrepreneurs are made to bear risk, while others

are given perfect insurance– and thus are less innovative. This enables them to reach a growth

rate between that implied by a fully cuddly reward structure and the higher growth rate of the

cutthroat reward structure.

Finally, for another range of parameters (part 3 of the proposition), there is “institutional”

and technology convergence in that followers also adopt cutthroat reward structures. When

this is the case, technology spillovers ensure not only the same long-run growth rate across all

countries but convergence in income and technology levels. In contrast, in other cases, countries

maintain their different institutions (reward structures), and as a result, they reach the same

growth rate, but their income levels do not converge.

The growth dynamics implied by this proposition are also interesting. These are shown

in Figures 6-8. Figure 6 corresponds to the part 1 of Proposition 5, and shows the pattern

where, starting with a low enough initial condition, i.e., nj (0) < m1/φ, cutthroat capitalism

is followed by cuddly capitalism. As the figure shows, when nj (t) reaches m1/φ, the rate of

convergence changes because there is a switch from cutthroat to cuddly capitalism. This figure

also illustrates another important aspect of Proposition 5: there is institutional divergence as a

country converges to the technological leader– and as a consequence of this, this convergence is

incomplete, i.e., nj (t) converges to (go/gc)1/φ. The figure also shows that countries that start

out with nj (0) > m1/φ will choose cuddly capitalism throughout.

Figure 7 shows the somewhat different pattern of convergence implied by part 2 of the propo-

sition, where followers reach the growth rate of the leader in finite time and at a higher level of

relative income– because they choose a mixed reward structure in the limit. Nevertheless, insti-

27

Figure 6: Growth dynamics: part 1 of Proposition 5

tutional and level differences between the leader and followers remain. In Figure 8 corresponding

to part 3, leaders and followers adopt the same institutions and there is complete convergence.

4.2 Time-Varying Reward Structures With Epstein-Zin Preferences

In this subsection, we show that a variant of Proposition 5 holds with the preferences given in

(8). Our main result in this subsection is presented in the next proposition, and because the

proof illustrates the growth dynamics in a complementary manner to Proposition 5, we provide

it in the text.

Proposition 6 Suppose country social planners maximize (8), the world technology frontier isgiven by (5), Assumptions 1 and 2

′, and Condition 1

′hold, and technology spillovers are large

in the sense that φ > 1− θ. Suppose also that either θ < λ < 1 or θ > λ > 1. Let

mo ≡(1− θ)ωo(gc − go) + (1− λ)go(ωo − ωc)

(ρ+ φgc)(ωo − ωc),

and

mc ≡(1− θ)ωc(gc − go) + (1− λ)gc(ωo − ωc)

(ρ+ φgc)(ωo − ωc).

Then the world equilibrium is characterized as follows:

1. Ifgogc> mo > mc, (27)

there exist m, m, T and T ′ such that for nj (0) < m1/φ, the reward structure of country j

is cutthroat (i.e., sj(t) = c or uj (t) = 1) for all t ≤ T , and then at t = T , we have nj (T ) =

m1/φ and country j adopts a “mixed”reward structure until T ′ (i.e., uj (t) ∈ (0, 1)) for all

28

Figure 7: Growth dynamics: part 2 of Proposition 5

T ′ > t > T . Then at t = T ′, we have nj (T ′) = m1/φ and country j switches to a cuddly

reward structure (i.e., sj(t) = o or uj (t) = 0) for all t ≥ T ′, and nj (t)→ (go/gc)1/φ.

2. Ifgogc< mo and 1 > mc, (28)

then there exist m and m such that for m1/φ < nj (0) < m1/φ, the reward structure of

country j is mixed (i.e., uj (t) ∈ (0, 1)) for all t, and (mj (t) , uj (t)) → (m∗, u∗). If

nj (0) < m1/φ, then country j first adopts a cutthroat reward structure (i.e., sj(t) = c or

uj (t) = 1) until some T , and then switches to a mixed reward structure, again converging

to a unique (m∗, u∗), and if nj (0) > m1/φ, then country j first adopts a cuddly reward

structure (i.e., sj(t) = o or uj (t) = 0) until some T ′, and then switches to a mixed reward

structure, again converging to a unique (m∗, u∗).

3. If

mo > mc > 1, (29)

then for any nj (0) < 1, the reward structure of country j is cutthroat for all t (i.e.,

sj(t) = c or uj (t) = 1 for all t).

Proof. Now the optimization problem for a follower country, dropping the j subscript, can be

written as

maxu(t)

∫∞0 e−ρtw(u(t))

1−λ1−θm(t)

1−λφ N`(t)

1−λdt s.t: m(t) = φ[g(u(t))− gcm(t)],

N`(t) = gcN`(t)m(0) = (N(0)/N`(0))φ given.

29

Figure 8: Growth dynamics: part 3 of Proposition 5

Here, u(t) ∈ [0, 1] is again the fraction of entrepreneurs receiving cutthroat incentives, and

m(t) ≡ (N (t) /N` (t))φ. Now substituting for N`, this is equivalent to:

maxu(t)

∫∞0 e−(ρ−(1−λ)gc)tw(u(t))

1−λ1−θm(t)

1−λφ dt s.t: m(t) = φ[g(u(t))− gcm(t)],

m(0) = (N(0)/N`(0))φ given.

The current value Hamiltonean for this problem can be written as

H = w(u(t))1−λ1−θm(t)

1−λφ + µ(t)φ[g(u(t))− gcm(t)],

which is strictly concave in m when φ > 1 − λ, and strictly concave in u when 1 > λ > θ or

θ > λ > 1.

Consider the candidate solution given by the Maximum Principle, i.e., as a solution to the

following equations:

∂H

∂u= Ψ(t) ≡ 1− λ

1− θ (ωc − ωo)ω (u (t))θ−λ1−θ m(t)

1−λφ + µ(t)φ(gc − go) = 0 for 0 ≤ u (t) ≤ 1

m(t) = φ[g(u(t))− gcm(t)]

µ(t) = (ρ− (1− λ)gc + φgc)µ(t)− 1− λφ

ω (u (t))1−λ1−θ m(t)

1−λφ−1,

together with the transversality condition, not taking the form

limt→∞

e−(ρ−(1−λ)gc)tµ(t) = 0.

If the first condition cannot be satisfied for interior u (t), we have a corner solution at 0 or 1. In

particular, substituting for u (t) = 0, we obtain the following curve in the (m,µ) space:

Ψ0(t) ≡ 1− λ1− θ (ωc − ωo)ω

θ−λ1−θo m(t)

1−λφ + µ(t)φ(gc − go) = 0,

30

Figure 9: Case 1– Cuddly Reward Structure in the Limit.

and substituting for u (t) = 1,

Ψ1(t) ≡ 1− λ1− θ (ωc − ωo)ω

θ−λ1−θc m(t)

1−λφ + µ(t)φ(gc − go) = 0.

To the right of the first curve shown in Figure 9, the candidate solution involves u (t) = 0, and

to the left of the second curve, u (t) = 1. In-between, the solution involves mixed rewards. The

corresponding rewards can be straightforwardly solved out for

u(t) = − 1

ωo − ωc

( (1− θ)µ(t)φ(gc − go)(1− λ)(ωo − ωc)m(t)

1−λφ

) 1−θθ−λ

− ωo

.Substituting this into the above differential equations for m (t) and µ (t), we obtain two

autonomous differential equations in these two variables describing the dynamics with mixed

rewards:

m(t) = φ

go − (gc − go)ωo − ωc

( (1− θ)µ(t)φ(gc − go)(1− λ)(ωo − ωc)m(t)

1−λφ

) 1−θθ−λ

− ωo

− gcm(t)

,µ(t) = (ρ− (1− λ)gc + φgc)µ(t)− 1− λ

φ

((1− θ)φ(gc − go)(1− λ)(ωo − ωc)

) 1−λθ−λ

µ(t)1−λθ−λm(t)

θ(1−φ)+λ−1φ(θ−1) .

When these differential equations intersect with the Ψ0(t) and Ψ1(t) curves, then u (t) takes the

value 0 or 1, respectively, and these two differential equations simplify to

m(t) = φ[go − gcm(t)], (30)

µ(t) = (ρ− (1− λ)gc + φgc)µ(t)− 1− λφ

ω1−λ1−θo m(t)

1−λφ−1

31

or to

m(t) = φ[gc − gcm(t)]

µ(t) = (ρ− (1− λ)gc + φgc)µ(t)− 1− λφ

ω1−λ1−θc m(t)

1−λφ−1.

The full system is drawn for case 1, where gogc> mo > mc, in Figure 9. In this case, it can

be verified that the intersection of the loci for m(t) = 0 and µ(t) = 0 is indeed at m = go/gc

(corresponding to n = (go/gc)1/φ). The laws of motion plotted in the figure also make it clear

that the candidate solution trajectory will start with sj(t) = c or uj (t) = 1 if m (0) < m

or n (0) < m1/φ, and then switch to a mixed reward structure (i.e., uj (t) ∈ (0, 1)), and then

at m = m or n = m1/φ, two switch to sj(t) = o or uj (t) = 0. Throughout m (t) increases

monotonically towards go/gc. To complete the proof of the first part, we only need to show that

this candidate solution is optimal for the country j planner (and then concavity, combined with

the Mangasarian suffi ciency condition as in the proof of Proposition 5 in the Appendix, ensures

that this is the unique optimal plan). This follows by integrating out m (t) and µ (t) using (30)

to obtain

Ψ(t) =1− λ1− θ (ωo − ωc)ω

θ−λ1−θo

∫ ∞t

e−(ρ−(1−λ)gc+φgc)τm(τ)1−λφ−1

(mo −m(τ))dτ

for m (t) ≥ m. If u(t) = 0 for all t, then we have m(t) → go/gc, ensuring that Ψ(0) ≤ 0. This

implies that the above plan is optimal for m (t) ≥ m. Now considering m (0) < m, only the path

shown in the figure avoids a jump and can be optimal. This completes the proof of case 1.

Next consider case 3. Given the parameter configuration, now the unique intersection of the

loci for m(t) = 0 and µ(t) = 0 is at m = 1 as shown in Figure 10. The laws of motion in this

case also show easily that starting with any m (0) < 1, the unique solution involves u(t) = 1

for all t, converging to m = 1. With the same argument invoking the Mangasarian suffi ciency

condition, this completes the proof of case 3.

Finally consider case 2. the intersection of the loci for m(t) = 0 and µ(t) = 0 is in the

interior, mixed rewards range. The laws of motion, depicted in Figure 11, again show that there

is a unique trajectory converging to this intersection, which involves mixed reward structure

with the fraction u∗ ∈ (0, 1) of entrepreneurs receiving cutthroat rewards. This completes the

proof of the proposition.

First note that the condition that θ < λ < 1 or θ > λ > 1 is now imposed to ensure concavity

(together with the condition from Proposition 5 that φ > 1 − θ). Second, this proposition

makes it clear that essentially all the substantive results from Proposition 5 generalize to an

economy with the more general Epstein-Zin preferences. Inspection shows that the three different

cases of Proposition 5 correspond to the three different cases of Proposition 6, and have the

same economic content implications. Third, however, both the mathematical argument for

establishing these results and the nature of the results are somewhat different. In particular,

the transition to mixed rewards in the third case happens smoothly rather than with the jump

as in Proposition 5.

32

Figure 10: Case 3– Cutthroat Reward Structure Everywhere.

Figure 11: Case 2– Mixed Reward Structure in the Limit.

33

4.3 General Convex Aggregators for World Technology Frontier

We next show that the main result of this section holds with general aggregators of the form

(6) provided that these aggregators are suffi ciently “convex,” i.e., putting suffi cient weight on

technologically more advanced countries. The main difference from the rest of our analysis

is that with such convex aggregators, the world growth rate is no longer determined by the

reward structure (and innovative activities) of a single technology leader, but by a weighted

average of all economies. Nevertheless, the same economic forces exhibit themselves because the

convexity of these aggregators implies that the impact on the world growth rate of a change in

the reward structure of a technologically advanced country would be much larger than that of

a backward economy, and this induces the relatively advanced economies to choose cutthroat

reward structures, while relatively backward countries can free-ride and choose cuddly reward

structures safe in the knowledge that their impact on the long-run growth rate of the world

economy (and thus their own growth rate) will be small.

Proposition 7 Suppose that country social planners maximize (7) or (8), and the world tech-nology frontier is given by (6). Then there exist σ < 0, ρ > 0 and γ < 1 such that when

σ ∈ (σ, 0), ρ ≤ ρ and γ > γ there is no symmetric world equilibrium with all countries choosing

the same reward structure. Instead, there exists T < ∞ such that for all t > T , a subset of

countries will choose a cutthroat reward structure while the remainder will choose a cuddly or

mixed reward structure.

Proof. See the Appendix.

5 Equilibrium under Domestic Political Constraints

In this section, we focus on the world economy with two countries, j and j′, and also simplify

the discussion by assuming that nj′ (0) = nj (0), by focusing on time-invariant reward structures

as in Section 3, and also by assuming that the world technology frontier is given by (5) again

as in Section 3. This implies that there are two asymmetric equilibria, one in which country

j is the technology leader and j′ the follower, and vice versa. We also suppose that the social

planner in country j is subject to domestic political constraints imposed by a labor movement or

a social democratic party, which prevent the ratio of rewards when successful and unsuccessful

to be more than some amount ζ. There are no domestic constraints in country j′. If ζ ≥ A,

then domestic constraints have no impact on the choice of country j, and there continue to be

two asymmetric equilibria.

Suppose instead that ζ < A. This implies that because of domestic political constraints, it

is impossible for country j to adopt a cutthroat strategy regardless of the strategy of country j′.

This implies that of the two asymmetric equilibria, the one in which country j adopts a cutthroat

reward structure disappears, and the unique equilibrium (with time-invariant strategies) becomes

the one in which country j′ adopts the cutthroat strategy and country j chooses an egalitarian

34

structure. However, from Proposition 5 above, this implies that country j will now have higher

welfare than in the other asymmetric equilibrium (which has now disappeared). This simple

example thus illustrates how domestic political constraints, particularly coming from the left and

restricting the amount of inequality in society, can create an advantage in the world economy.

We next show that this result generalizes to the case in which the two countries do not start

with the same initial level of technology. To do this, we relax our focus on equilibria in which

the leader at time t = 0 always remains the leader. Let us also suppose, without loss of any

generality, that country j′ is technologically more advanced at t = 0, so that nj (0) ≤ 1. Then

we have the following proposition.

Proposition 8 Suppose that country social planners maximize (7) and the world technologyfrontier is given by (5). Suppose also that there are two countries j and j′ with initial technology

levels Nj′ (0) ≥ Nj (0) (which is without loss of any generality), they are restricted to time-

invariant reward structures, Condition 1 and (16) hold.

1. There exists δ > 0 such that for all nj (0) > 1−δ, there are two asymmetric time-invariantequilibria, one in which country j adopts a cutthroat reward structure and country j′ adopts

a cuddly reward structure, and vice versa.

2. If domestic constraints imply that country j cannot adopt a cutthroat reward structure,

then the unique time-invariant equilibrium is the one in which country j′ adopts a cutthroat

reward structure and country j adopts a cuddly reward structure. The equilibrium welfare

of country j is greater than that of country j′.

Proof. The first part follows by noting that when Condition 1 holds and the gap between thetwo countries is small (i.e., nj (0) > 1 − δ), then it also ensures that whichever country willdetermine the rate of growth of the world technology in the near future prefers to choose the

cutthroat reward structure (and again for nj (0) > 1 − δ, there exists T such that this countrydetermines the world growth rate for t > T ). Part 2 then immediately follows from Proposition

5.

An interesting implication of this result is that country j, which has a stronger labor move-

ment or social democratic party, benefits in welfare terms by having both equality and rapid

growth, but in some sense exports its potential labor conflict to country j′, which now has to

choose a reward structure with significantly greater inequality.

In the context of the comparison of the US to Scandinavian economies, the latter clearly have

a history of stronger labor movement and social democratic party, suggesting that this might

have been one of the factors influencing the specific pattern of asymmetric world equilibrium

that has developed over the last several decades. Friedman (2010) provides an overview of

existing cross-national historical data on union density which shows that in 1928, just prior

to the date when the Swedish Social Democrats took power (1932), its unionization rate was

32.0%. In Denmark, this was 39.7% and in Norway 17.4%. In the US this number was 9.9% in

35

1928. It is not just the extent of unionization but how it is organized. US Unions like those of

Britain tended to be along craft lines with multiple unions in a single firm, making it much more

diffi cult for the labor movement to act collectively. In Scandinavia unions were organized by

industry and were much more encompassing. Finally, the rise of social democratic parties which

cemented the Scandinavian model into place had its roots in the late 19th century (see Lundberg

and Åmark, 2001, for Sweden, Baldwin, 1992, more generally) and its particular universalistic

and tax financed nature was a result of what Gourevitch (1986) calls the “Red-Green Coalition”

which linked poor rural peasants with urban industrial workers (see also Baldwin, 1992). In the

US similar debates took place and the coalitions that formed within the Populist and Progressive

movements had Red-Green elements. Though they also managed to push progressive reforms,

such as the introduction of the income tax in 1913, these movements were weaker and failed to

unite with a factious labor movement. While the 1920s saw left-wing political parties come to

power in Denmark (the Social Democrats came to power in 1924) and Norway (Norwegian Labor

Party formed its first government in 1928), left-wing US political parties severely declined. Thus

just at the critical juncture where many of the institutions of 20th century developed countries

states were formed, the strength of labor movements and left-wing political parties was much

greater in Scandinavia than in the US.

6 Conclusion

In this paper, we have taken a first step towards a systematic investigation of institutional

choices in an interdependent world– where countries trade or create knowledge spillovers on

each other. Focusing on a model in which all countries benefit and potentially contribute to

advances in the world technology frontier, we have suggested that the world equilibrium may

necessarily be asymmetric. In our model economy, because effort by entrepreneurs is private

information, a greater gap of incomes between successful and unsuccessful entrepreneurs– thus

greater inequality– increases innovative effort and a country’s contributions to the world tech-

nology frontier. Under plausible assumptions, in particular with suffi cient risk aversion and

a suffi cient return to entrepreneurial effort, some countries will opt for a type of “cutthroat”

capitalism that generates greater inequality and more innovation and will become the technol-

ogy leaders, while others will free-ride on the cutthroat incentives of the leaders and choose a

more “cuddly”form of capitalism. We have also shown that, paradoxically, starting with similar

initial conditions, those that choose cuddly capitalism, though poorer, will be better off than

those opting for cutthroat capitalism. Nevertheless, this configuration is an equilibrium because

cutthroat capitalists cannot switch to cuddly capitalism without having a large impact on world

growth, which would ultimately reduce their own welfare. This perspective therefore suggests

that the diversity of institutions we observe among relatively advanced countries, ranging from

greater inequality and risk taking in the United States to the more egalitarian societies sup-

ported by a strong safety net in Scandinavia, rather than reflecting differences in fundamentals

between the citizens of these societies, may emerge as a mutually self-reinforcing equilibrium. If

36

so, in this equilibrium, we cannot all be like the Scandinavians, because Scandinavian capitalism

depends in part on the knowledge spillovers created by the more cutthroat American capitalism.

Clearly, the ideas developed in this paper are speculative. We have theoretically shown

that a specific type of asymmetric equilibrium emerges in the context of a canonical model of

growth– with knowledge spillovers combined with moral hazard on the part of entrepreneurs.

Whether these ideas contribute to the actual divergent institutional choices among relatively

advanced nations is largely an empirical question. We hope that our paper will be an impetus

for a detailed empirical study of these issues.

In addition, there are other interesting theoretical questions raised by our investigation.

Similar institutional feedbacks may also emerge when countries interact via international trade

rather than knowledge spillovers. For example, if different stages of production require different

types of incentives, specialization in production resulting in a Ricardian equilibrium may also

lead to “institutional specialization”. In addition, while we have focused on a specific and

simple aspect of institutions, the reward structure for entrepreneurs, our results already hint

that there may be clusters of institutional characteristics that co-vary– for example, strong

social democratic parties and labor movements leading to cuddly capitalism domestically and

to cutthroat capitalism abroad. Institutional choices concerning educational systems, labor

mobility, and training investments may also interact with those related to reward structures for

entrepreneurs and workers. We believe that these are interesting topics for future study.

References

Acemoglu, Daron (2009) Introduction to Modern Economic Growth, Princeton UniversityPress.

Acemoglu, Daron, Philippe Aghion and Fabrizio Zilibotti (2006) “Distance toFrontier, Selection, and Economic Growth,” Journal of the European Economic Association,

MIT Press, vol. 4(1), pages 37-74, 03.

Acemoglu, Daron and Jorn-Steffen Pischke (1998) “Why Do Firms Train? Theoryand Evidence”Quarterly Journal of Economics, 113:1, 78-118.

Acemoglu, Daron and Jaume Ventura (2001) “The World Income Distribution”Quar-terly Journal of Economics, 117:2, 659-694.

Akkermans, Dirk, Carolina Castaldi, and Bart Los (2009) “Do ‘liberal marketeconomies’really innovate more radically than ‘coordinated market economies’? Hall and Sos-

kice reconsidered,”Research Policy, 38, 181-91.

Alesina, Alberto and Edward L. Glaeser (2004) Fighting Poverty in the US andEurope: A World of Difference, New York: Oxford University Press.

Alesina, Alberto, Edward L. Glaeser and Bruce Sacerdote (2001) “Why Doesn’t theUnited States Have a European-Style Welfare State?”Brookings Paper on Economics Activity,

Fall 2001, 187-278.

37

Alesina, Alberto, Edward L. Glaeser and Bruce Sacerdote (2005) “Work andLeisure in the U.S. and Europe: Why So Different?”NBER Macroeconomic Annual, 1-64.

Alesina, Alberto and with Eliana La Ferrara (2005) “Preferences for Redistributionin the Land of Opportunities,”Journal of Public Economics, 89, 897-931.

Algan, Yann, Pierre Cahuc and Marc Sangnier (2011) “Effi cient and Ineffi cientWelfare States,”https://sites.google.com/site/pierrecahuc/ unpublished_papers.

Atkinson, Anthony B., Thomas Piketty and Emmanuel Saez (2007) “Top Incomesin the Long Run of History”Journal of Economic Literature, 49:1, 3-71.

Baldwin, Peter (1992) The Politics of Social Solidarity: Class Bases of the EuropeanWelfare State, 1875-1975, New York: Cambridge University Press.

Bénabou, Roland J.M. (2000) “Unequal Societies: Income Distribution and the SocialContract,”American Economic Review, 90, 96-129.

Bénabou, Roland J.M. and Efe Ok (2001) “Social Mobility and the Demand for Re-distribution: The POUM Hypothesis,”Quarterly Journal of Economics, 116:2, 447-487.

Bénabou, Roland J.M. and Jean Tirole (2006) “Belief in a Just World and Redistrib-utive Politics,”Quarterly Journal of Economics, 121:2, 699-746.

Blanchard, Olivier (2004) “The Economic Future of Europe,”Journal of Economic Per-spectives, 18:4, 3-26.

Bottazzi, Laura and Giovanni Peri (2003) “Innovation and spillovers in regions: Evi-dence from European patent data,”European Economic Review, 47:4, 687-710.

Cameron, David R. (1978) “The Expansion of the Public Economy: A Comparative

Analysis,”American Political Science Review, 72, 1243—61.

Cardoso, Fernando H. and Enzi Faletto Enzo (1979) Dependency and Developmentin Latin America, Berkeley: University of California Press.

Coe, David and Elhanan Helpman (1995) “International R&D spillovers,”European

Economic Review vol. 39(5), pages 859-887, May.

Crouch, Colin (2009) “Typologies of Capitalism,”in Bob Hancké ed. Debating Varietiesof Capitalism: A Reader, New York: Oxford University Press.

Davis, Donald R. (1998) “Does European Unemployment Prop Up American Wages?National Labor Markets and Global Trade,”American Economic Review, 88(3), 478-494.

Epstein, Larry G. and Stanley E. Zin (1989) “Substitution, Risk Aversion and Tem-poral Behavior of Consumption and Asset Returns: A Theoretical Framework“ Econometrica,

57:4, 937-969.

Esping-Anderson, Gösta (1990) The Three Worlds of Welfare Capitalism, Princeton:Princeton University Press.

Friedman, Gerald (2010) “Labor Unions in the United States,”

http://eh.net/encyclopedia/article/friedman.unions.us

Gerschenkron, Alexander (1962) Economic backwardness in historical perspective, abook of essays, Belknap Press of Harvard University Press, Cambridge, Massachusetts.

38

Griffi th, Rachel, Stephen Redding and John Van Reenen (2003) “R&D and Ab-

sorptive Capacity: Theory and Empirical Evidence,”Scandinavian Journal of Economics, Wiley

Blackwell, vol. 105(1), pages 99-118, 03.

Gourevitch, Peter (1986) Politics in Hard Times, Ithaca: Cornell University Press.Hall, Bronwyn, Adam Jaffe, and Manuel Trajtenberg (2001) “The NBER Patent

Citation Data File: Lessons, Insights and Methodological Tools,”NBER Working Paper 8498.

Hall, Peter and David Soskice (2001) Varieties of Capitalism: The Institutional Foun-dations of Comparative Advantage, Oxford University Press, USA.

Hölmstrom, Bengt (1979) “Moral Hazard and Observability,”The Bell Journal of Eco-nomics, Vol. 10, No. 1 , pp. 74-91.

Howitt, Peter (2000) “Endogenous Growth and Cross-Country Income Differences,”American Economic Review, 90:4, 829-846.

Keller, Wolfgang (2001) “Knowledge Spillovers at the World’s Technology Frontier,”CEPR Discussion Papers 2815.

Kerr, William (2008) “Ethnic Scientific Communities and International Technology Dif-fusion,”Review of Economics and Statistics, 90:3, 518-537

Krueger, Dirk and Krishna Kumar (2004) “US-Europe differences in technology-drivengrowth: quantifying the role of education,”Journal of Monetary Economics, 51:1, 161-190.

Krugman, Paul and Anthony Venables (1996) “Integration, specialization, and ad-justment,”European Economic Review, 40, 959-967.

Landier, Agustin (2005) “Entrepreneurship and the Stigma of Failure,”mimeo.Lundberg, Urban and Klas Åmark (2001) “Social Rights and Social Security: The

Swedish Welfare State, 1900—2000,”Scandinavian Journal of History, 26:3, 157-176.

Matsuyama, Kiminori (2002) “Explaining Diversity: Symmetry-Breaking in Comple-mentarity Games,”American Economic Review, 92, 241-246.

Matsuyama, Kiminori (2005) “Symmetry-Breaking,”in L. Blume and S. Durlauf, eds.,the New Palgrave Dictionary of Economics, 2nd Edition, Palgrave Macmillan.

Matsuyama, Kiminori (2007) “Aggregate Implications of Credit Market Imperfections,”in D. Acemoglu, K. Rogoff, and M. Woodford, eds., NBER Macroeconomics Annual 2007.

Milesi-Ferretti, Gian Maria, Roberto Perotti and Massimo Rostagno (2002)“Electoral Systems and Public Spending,”Quarterly Journal of Economics,

Moene, Karl Ove and Michael Wallerstein (1997) “Pay Inequality,”Journal of LaborEconomics, 15(3), 403-430.

Nelson, Richard and Edmond Phelps (1966) “Investment in Humans, TechnologicalDiffusion, and Economic Growth,”The American Economic Review, 56:1/2 69-75.

OECD (2010) Hours Worked: Average annual hours actually worked, OECD Employmentand Labour Market Statistics (database).

OECD (2011) OECD Statistics.Persson, Torsten and Guido Tabellini (2003) The Economic Effect of Constitutions,

39

Cambridge: MIT Press.

Piketty, Thomas (1995) “Social mobility and redistributive politics,”Quarterly journalof economics, 110(3), 551-584.

Rodrik, Dani (2008) “Second-Best Institutions,”American Economic Review, 98(2), 100-104.

Romer, Paul (1990) “Endogenous Technological Change,”Journal of Political Economy,98:5, pp. S71-S102.

Saint-Paul, Gilles and Thierry Verdier (1993) “Education, democracy and growth,”Journal of Development Economics, 42, 399-407.

Schor, Juliet (1993) The Overworked American: The Unexpected Decline of Leisure, BasicBooks.

Smeeding, Timothy (2002) “Globalization, Inequality, and the Rich Countries of the G-20: Evidence from the Luxembourg Income Study (LIS),”Center for Policy Research Working

Papers 48, Maxwell School, Syracuse University.

Taylor, Mark Z. (2004) “Empirical Evidence Against Varieties of Capitalism’s Theory ofTechnological Innovation,”International Organization, 58, 601-631.

Wallerstein, Immanuel (1974-2011) The Modern World System, 4 Volumes, New York:Academic Press.

WIPO, Economics and Statistics Division (2011)World Intellectual Property Indica-tors, World Intellectual Property Organization.

40

Appendix

Derivation of Equation (12) . To derive (12), we need to characterize the equilibrium prices

and quantities in country j as a function of current technology Nj (t). This follows directly from

Chapter 18 of Acemoglu (2009). Here it suffi ces to note that the final good production function

(1) implies iso-elastic demand for machines with elasticity 1/β, and thus each monopolist will

charge a constant monopoly price of ψ/ (1− β), where recall that ψ is the marginal cost in terms

of the final good of producing any of the machines given its blueprint (invented or adapted from

the world technology frontier). Our normalization that ψ ≡ 1 − β then implies that monopolyprices and equilibrium quantities are given by pxj (ν, t) = 1 and xj(ν, t) = Lj = 1 for all j, ν and t.

This gives that total expenditure on machines in country j at time t will beXj(t) = (1− β)Nj(t),

while total gross output is

Yj(t) =1

1− βNj(t).

Therefore, total net output, left over for distributing across all workers/entrepreneurs is

NYj (t) ≡ Yj (t)−Xj (t) = BNj (t), where

B ≡ β (2− β)

1− β ,

which gives us (12).

Proof of Corollary 2.

Part 1: It is straightforward to verify that(ωcωo

) 11−θ

is decreasing in A (defined in (9))

and γ. Its dependence on θ is more complicated. As noted in the text, it is decreasing in A.

Differentiation and algebra then establishes A is increasing in θ when γ ≤ γ ≡ 1 −√

q0(1−q0)q1(1−q1) .

Also,(ωcωo

) 11−θ

is decreasing in θ for fixed A. Therefore,(ωcωo

) 11−θ

is decreasing in θ when γ ≤ γ.

Moreover, defining θmax ≡ 1−log(

1−q01−q1

)

log(1−γ) > 1, we also have limθ→θmax

(ωcωo

) 11−θ

= 0. Thus there

exists θ∗(φ, γ) ∈ (0, θmax(γ)) such that when θ ≥ θ∗(φ, γ),(ωcωo

) 11−θ

<(gogc

) 1φ, and moreover θ ≥

θ∗(φ, γ) is decreasing in γ and φ (the latter from the fact that the right-hand side of inequality

is increasing in φ.

Part 2: When θ < 1, Condition 1 requires that

ωcωo

>ρ− (1− θ)gcρ− (1− θ)go

.

Since ωcωo< 1 and (1− θ)gc > (1− θ)go, there exists a unique ρ(θ, γ) ∈ [(1− θ)gc,∞) such that

is inequality satisfied if and only if ρ < ρ(θ, γ). When θ > 1, ωc < ωo < 0, and Condition 1

requiresωcωo

<ρ− (1− θ)gcρ− (1− θ)go

.

In this case, some algebra establishes that the same conclusion follows provided that γ ≤ γ ≡1−

√q0(1−q0)q1(1−q1) and θ ∈

[1, θ(γ)

]where θ(γ) > 1. Moreover, in both cases ρ(θ, γ) is decreasing in

θ and γ.

41

Proof of Proposition 5. We rewrite (22) with a change of variable for mj ≡ (Nj/N`)φ ≤ 1

as:

Wj(mj (t)) = N` (t) maxu(.)∈[0,1]

∫ ∞t

e−(ρ−(1−θ)gc)(τ−t)ω (u(τ))mj (τ)1−θφ dτ (31)

·mj (τ) = φ [g (u (τ))− gcmj (τ))] .

The solution to this problem would be the “closed loop” best response of follower j to the

evolution of the world technology frontier driven by the technology leader, `. The Markov

perfect equilibrium corresponds to the situation in which all countries use “open loop”strategies.

However, given our focus on equilibria in which the same country, `, remains the leader and

adopts a cutthroat reward structure (under Condition 1), the open loop and the closed loop

solutions coincide, because under this scenario, country ` always adopts a cutthroat reward

structure, regardless of the strategies of other countries. Hence we can characterize the equilibria

by deriving the solution to (31).

We now proceed by defining the current-value Hamiltonian, suppressing the country index

j to simplify notation,

H(m (t) , u (t) , µ (t)) = ω(u (t))m (t)1−θφ + µ (t)φ [g(u (t))− gcm (t)] ,

where µ(t) is the current-value co-state variable. We next apply the Maximum Principle to

obtain a candidate solution. This implies for the control variable (reward structure) u (t) the

following bang-bang form:

u (t)

= 1∈ [0, 1]

= 0if

Ψ (t) < 0Ψ (t) = 0Ψ (t) > 0

(32)

where Ψ (t) is the switching function:

Ψ (t) ≡ (ωo − ωc)m (t)1−θφ − µ (t)φ [gc − go] . (33)

In addition,

·m (t) = φ [g(u (t))− gcm (t)] with m (0) > 0 given (34)·µ (t) = (ρ− (1− θ)gc + φgc)µ (t)− 1− θ

φm (t)

1−θφ−1ω (u (t)) ,

and the transversality condition,

limt→∞

e−(ρ−(1−θ)gc)tµ(t) = 0. (35)

Now combining (33) with (34), we have

Ψ(t) = (ρ− (1− θ)gc + φgc) Ψ(t) + (ωo − ωc)(ρ+ φgc)m(t)1−θφ−1

(m−m(t)) , (36)

where m is given by (23) in the statement of Proposition 5. Integrating (36), we obtain

Ψ(t) = (ωo − ωc)(ρ+ φgc)

∫ ∞t

e−((ρ−(1−θ)gc+φgc))(τ−t)m(τ)1−θφ−1

(m(τ)− m) dτ. (37)

42

Moreover, (32) implies that in the candidate solution, cutthroat (cuddly) reward structures

will be adopted at time t when Ψ (t) < 0 (> 0). Notice first that (34) implies that

·m (t) ≥ φ [go − gcm (t)]

Thus,

m(t) ≥ gogc

+ (m(0)− gogc

)e−φgct (38)

Next observe the following about the candidate solution.

1. Suppose m < go/gc (corresponding to part 1 of Proposition 5). One can first notice that

the control variable u (t) can only take the extreme values 0 or 1. To see this, suppose to obtain

a contradiction that in some interval t ∈ [t1, t2], u(t) ∈ (0, 1). Then also Ψ (t) = 0 on that same

interval [t1, t2]. Therefore for t ∈ [t1, t2], Ψ (t) = 0, but then (36) implies that m(t) is a constant

equal to m. Thus·m (t) = 0 and u(t) = u = (gcm− go) /(gc−go), which together with m < go/gc

implies that u < 0, yielding a contradiction.

Next consider the following cases:

• If m (0) > go/gc, Then (38) implies that m(t) ≥ go/gc > m for all t. Hence (37) implies

that Ψ (t) > 0 for all t, and thus u (t) = 0 for all t (which also implies from (34) that m (t)

is monotonically decreasing towards go/gc).

• If m (0) < go/gc, then (38) implies that lim inf m (t) ≥ go/gc > m, and thus lim inf Ψ (t) >

0. Hence there exists T ′ such that for t > T ′, Ψ (t) > 0, and thus u (t) = 0. Two cases

need to be considered:

- Case i) For all t ∈ [0, T ′], Ψ (t) > 0 and therefore for all t ≥ 0, u (t) = 0 (and m (t) is

monotonically increasing towards go/gc).

- Case ii) There exists t′ ∈ [0, T ′] such that Ψ (t′) = 0. Let to < T be the maximum of such

dates t′. We have Ψ(to) = 0. By definition of to, for all t > to Ψ(t) > 0. Hence we also have

Ψ′(to) > 0. Equation (36) implies that m(to) < m. Suppose now that there is another date

t” < to such that Ψ(t”) = 0 and take the largest of such dates t1 < to. By construction,

Ψ′ (t1) < 0. Also given that for all t ≥ to Ψ(t) ≥ 0, and the continuity of Ψ(t), Ψ(t) < 0 on

the interval t ∈ (t1, to) and Ψ(to) = Ψ(t1) = 0. Hence for t ∈ (t1, to), we also have u(t) = 1

and m(t) increasing in t (from (34)). It follows that m(t1) < m(to) < m. However, (36),

Ψ′ (t1) < 0 and Ψ(t1) = 0 jointly imply that m(t1) > m, yielding a contradiction Hence

there cannot exist another date t” < to such that Ψ(t”) = 0. Hence the function Ψ (t)

cannot change sign on [0, to). Given that at to Ψ(to) = 0, one should have Ψ(t) < 0 on

[0, to).

- From the previous discussion, it follows that there exists at most one date T ≥ 0 at which

the function Ψ (t) changes sign. When such a date T exists, it must be that Ψ (t) < 0 and

u(t) = 1 for t ∈ [0, T ) and Ψ (t) > 0 and u(t) = 0 for t ∈ (T,∞) . The existence of such

time T depends on the sign of Ψ (0). When Ψ (0) < 0, there exists such switching date

43

T at which Ψ (t) changes signs (< 0 to > 0). When conversely Ψ (0) > 0 the switching

function is positive for all t. Note also that m (t) is increasing in m (0). Now consider the

case where θ > 1. Then m < 0, and from (37), Ψ (0) > 0, so that there is no switching.

Next consider the case where θ < 1. Then m > 0, and this together with the condition

φ > 1− θ gives

∂Ψ (0)

∂m(0)= (ωo − ωc)(ρ+ φgc)

×∫ ∞

0e−((ρ−(1−θ)gc+φgc))τ ∂m(τ)

∂m(0)

[1− θφ

m(τ)−(

1− θφ− 1

)m

]m(τ)

1−θφ−2dτ > 0.

Hence Ψ (0) is increasing inm (0), and thus there existsm such that Ψ (0) ≷ 0 ifm (0) ≷ m

(i.e., n (0) ≷ m1/φ).

2. Suppose 1 > m > go/gc (corresponding to part 2 of Proposition 5). Then the following

choice of rewards structure satisfies (32):

u (t) =

0 if m (t) > mu∗ if m (t) = m1 if m (t) < m

where u∗ is such that m = g(u∗)/gc, and when m (t) = m, we have·m (t) = 0 and Ψ (t) = 0,

ensuring that this choice of reward structure does indeed satisfy (32). Note also that in this

case whenever m (t) > m (m (t) < m) m (t) declines (increases) to m monotonically, and at

m (t) = m, it remains constant.

3. Suppose m > 1 (corresponding to part 3 of Proposition 5). In this case, Ψ (t) < 0 for all

t (regardless of initial conditions), and thus u (t) = 1 for all t. Given this reward structure, in

this case m (t) monotonically converges to 1.

Finally, in each case, the candidate solution satisfies the transversality condition (35), and

the assumption that 1 − θ < φ ensures that Mangasarian’s suffi ciency condition is satisfied

(e.g., Acemoglu, 2009, Chapter 7). Thus the candidate solution characterized above is indeed a

solution and is unique. This completes the proof of Proposition 5.

Proof of Proposition 7. We will prove that under the hypotheses of the proposition, there

does not exist a symmetric equilibrium. We focus on the case in which preferences are given by

(7). The proof for the case in which they are given by (8) is similar.

Suppose first that all countries choose a cuddly reward structure for all t ≥ 0. Then the

world economy converges to a Balanced Growth Path (BGP) where every country has the same

level of income, Nj (t) / (1− β) = N (t) / (1− β), and grows at the same rate, which from (6) is

equal to·N(t)/N(t) = go. The time t welfare of country j in this equilibrium can be written as

Woj (t) =

∫ ∞t

e−δ(τ−t)ωo

(Nj (τ)

N(τ)

)1−θN(τ)1−θdτ,

44

which implies that for any ε > 0, there exists T1 such that for all t > T1, we are close enough to

the steady state equilibrium in the sense that 1− ε < Nj(t)N(t) < 1 + ε,

·N/N < go + ε, and

Woj (t) <

ωoN(t)1−θ(1 + ε)1−θ

ρ− (1− θ)(go + ε)

Consider now a deviation of one country k to a cutthroat reward structure at all times t > T1.

Denote by Nj(t), the new growth path of country j and by N(t) the growth path to the world

technology frontier. The world economy converges again to a new BGP with growth rate g.

This BGP growth rate can be written as

g =1

Jφ

1+φ

[(J − 1)g

1φσ−1σ

o + g1φσ−1σ

c

] σσ−1

φ1+φ

> go.

After this deviation, we have Nk(t) > Nk(t) and Nk(t) > N(t) for all t > T1. Then for ε1 > 0,

there exists T ′1 > T1 and ε′1 such that for all t > T ′1,·Nk/Nk ≥ g − ε1, and welfare of country k

satisfies

Wck(T1) =

∫ ∞T1

e−ρ(t−T1)ωcNk(t)1−θdt

=

∫ T ′1

T1

e−ρ(t−T1)ωcNk(t)1−θdt+ e−ρ(T ′1−T1)

∫ ∞T ′1

e−ρ(t−T ′1)ωcNk(t)1−θdt

> e−ρ(T ′1−T1)ωcNk(T

′1)1−θ

ρ− (1− θ)(g − ε′) .

Now using the fact that Nk(T′1) ≥ Nk(T

′1) ≥ ego(T

′1−T1)Nk(T1), a suffi cient condition for the

deviation for country k to be profitable is

e−(ρ−(1−θ)go)(T ′1−T1)ωcNk(T1)1−θ

ρ− (1− θ)(g − ε1)> ωo

Nk(T1)1−θ(1 + ε)1−θ

ρ− (1− θ)(go + ε)

> Wok(T1) =

∫ ∞T1

e−(ρ−(1−θ)go)(t−T1)ωoNk (t)1−θ dt.

Rearranging terms, this can be written as(ωcωo

) 11−θ

> (1 + ε) eρ−(1−θ)go

1−θ (T ′1−T1)

(ρ− (1− θ)(g − ε1)

ρ− (1− θ)(g0 + ε)

) 11−θ

. (39)

Next suppose that all countries adopt a cutthroat reward structure for all t ≥ 0. In this case,

the world economy converges to a BGP where every country has the same level of income and

grows at the same rate, which from (6) is equal to·N(t)/Nj(t) = gc. With a similar reasoning,

for ε > 0, there exists T2 such that for all j and t > T2, 1 − ε < Nj (t) /N(t) < 1 + ε and·N/N < gc + ε. Thus

Wcj (t) <

ωcN(t)1−θ(1 + ε)1−θ

ρ− (1− θ)(gc + ε)

45

Consider now a deviation of one country k to a cuddly reward structure at all time t > T2 while

all other countries j 6= k stay with cutthroat reward structures throughout. Denote the path of

technology of country j after this deviation by Nj(t), and the path of world technology frontier

by N (t). Clearly,·N(t)/ Nj(t) = g < gc, and moreover Nk(t) ≤ Nk (t) for all t > T2. Let us also

note that

g =

·N j(t)

N(t)=

1

J

[(J − 1)gc

σ−1σ + g

σ−1σ

o

] σσ−1

> go.

Now, again fixing ε2 > 0, there exists T ′2 > T2 such that for all t > T ′2,·Nk/Nk ≥ g− ε2, and the

welfare of country k satisfies

Wok(T2) =

∫ ∞T2

e−ρ(t−T )ωoNk(t)1−θdt

=

∫ T ′2

T2

e−ρ(t−T2)ωoNk(t)1−θdt+ e−ρ(T ′2−T2)

∫ ∞T ′2

e−ρ(t−T”2)ωoNk(t)1−θdt

> ωoNk(T2)1−θ∫ T ′2

T2

e−ρ(t−T )e(1−θ)go(t−T )dt+ e−ρ(T ′2−T2)ωoNk(T2)1−θe(1−θ)g0(T ′2−T2)

ρ− (1− θ)(g − ε2)

> ωoNk(T2)1−θ 1− e−(ρ−(1−θ)go)(T ′2−T2)

ρ− (1− θ) go+ e−(ρ−(1−θ)go)(T ′2−T2)ωo

Nk(T2)1−θ

ρ− (1− θ)(g − ε2),

where the second line uses the fact Nk(t) > Nk(T2)eg0(t−T2). Then a suffi cient condition for the

deviation to the cuddly reward structure for country k to be profitable is

e−(ρ−(1−θ)go)(T ′2−T2)ωoNk(T2)1−θ

ρ− (1− θ)(g − ε2)> ωc

N(T2)1−θ(1 + ε)1−θ

ρ− (1− θ)(gc + ε).

Since Nk(T2) > N(T2)(1− ε), this suffi cient condition can be rewritten as

1− ε1 + ε

(ρ− (1− θ)(gc + ε))

ρ− (1− θ)(g − ε2)

) 11−θ

e−ρ−(1−θ)go

1−θ (T ′2−T2) >

(ωcωo

) 11−θ

. (40)

Thus combining (39) and (40), we obtain that the following is a suffi cient condition for an

asymmetric equilibrium not to exist after some time T = max {T1, T2}:

1− ε1 + ε

(ρ− (1− θ)(gc + ε))

ρ− (1− θ)(g − ε2)

) 11−θ

e−ρ−(1−θ)go

1−θ (T ′2−T2) >

(ωcωo

) 11−θ

>1− ε1 + ε

eρ

1−θ (T ′1−T1)

(ρ− (1− θ)(g − ε1)

ρ− (1− θ)(go + ε)

) 11−θ

.

(41)

Now note that as σ ↑ 0 in (6), g −→ gc and g −→ gc. Therefore, for ε′ > 0, there exists

σ < 0 such that for σ > σ, g− ε′ < go and g− ε′ < go. Thus choosing ε, ε1, ε2, and ε′ suffi ciently

small, the following is also a suffi cient condition:

e−ρ−(1−θ)go

1−θ (T ′2−T2) >

(ωcωo

) 11−θ

> eρ−(1−θ)go(T ′1−T1)

1−θ

(ρ− (1− θ)gcρ− (1− θ)go

) 11−θ

. (42)

Finally, choosing ρ suffi ciently close to (1− θ) gc and defining T ≡ {T ′1 − T1, T′2 − T2}, a further

suffi cient condition is obtained as

e−(gc−go)T >

(ωcωo

) 11−θ

> e(gc−go)T(ρ− (1− θ)gcρ− (1− θ)go

) 11−θ

. (43)

46

For given choices of ε and ε1, T is fixed. Hence there exists ρ > (1− θ) gc such that for(1− θ) gc < ρ < ρ, the right-hand side term inequality is close to zero and the left-hand term is

given by some positive number. Next recall that

(ωcωo

) 11−θ

=

(q1A

1−θ + (1− q1)1

1−θ)

(1− γ)

q1A+ (1− q1).

When θ < 1, this tends to 0 as γ → 1 −(q0q1

)1/(1−θ). When θ > 1, this tends to 0 as γ →

1 −(

1−q01−q1

)1/(1−θ). Thus in both cases (for a fixed value of θ) their exists γ < 1 such that for

γ > γ,(ωcωo

) 11−θ

is sandwiched between these two terms, ensuring that (43) is satisfied and a

symmetric equilibrium does not exist.

Finally, when these conditions are satisfied, a similar analysis to that in the proof of Propo-

sition 5 implies that the equilibrium will take the form where after some T , subset of countries

choose a cuddly reward structure and the remaindered choose a cutthroat reward structure.

47